Hooboy. This may be prohibitive to bring to full code, so let me discuss some options to get the most safety increase for the least buck.
Romex NM-B is not legal for direct-burial or for running in conduit. But what really has me spooked is bootlegging a neutral off that ground wire. If anything breaks in that ground wire, it will energize every ground in your electrical system at 120VAC - touching a conduit, the panel, light switch screws, all the things that are supposed to be safe will be dangerous! Far away from the house you could be dead before they find you! Do not leave this situation to fester merely because bringing it all the way to current code would be hard: Don't let "perfect" be the enemy of the "good". Every option here restores the proper function of the safety ground and separates neutral from it.
Ed Beal discusses an issue with the ground rod.
I am willing to assume some defects may be grandfathered or permitted via local exemption.
Option 1: Go 120V-only
If you can bear having only 120V, then remove all markings from the white wire. In the main panel, black goes to a 120V breaker (or one side of a 240V breaker), white to neutral bar, and bare to ground bar (they may be the same in the main panel). In the sub-panel, black wire goes to one phase, white to the neutral bar, bare to the ground bar which are isolated. Every other row of breakers will not work - just move your needed breakers to rows that do work.
Cost: $30ish per building, for a separate neutral bar for the sub-panel
Option 2: Go 240V-only - temporary transformers for 120V
If your primary loads are 240V-only and/or multi-voltage loads (lighting, some gadgets), and your 120V loads can be hooked up temporarily as needed, easy - just reconfigure for 240V-only. Stop using neutral; the bare wire is ground-only.
Mark the white wire with tape on both ends to designate it a "hot". In the main panel, black and remarked-white to a 240V breaker. Bare to the ground bus. In the sub-panel, black and remarked-white to opposite "hots", bare to the ground bar. No neutral bar - no neutrals will exist in this panel! No exceptions! Every white wire in wiring will be marked with tape as it's not a neutral.
To power 240V loads, done. Many machines will run on 120-240V. Lighting particularly - check your fluorescent lights, the newest ballasts are 120-277V automatic switching, if yours isn't, just change ballast, they're $10-20. Good time to upgrade to T8 bulbs while you're at it. Many LED bulbs are 120-277V, I've been replacing 175W barn lights with 15W LED lights that are $10 and multivoltage. Old mercury, sodium and halide lights have transformers, and they are often jumpable for 240V. For plug-connected devices that can accept 240V, change your wired receptacles and plugs to NEMA 6-15 or 6-20.
To temporarily power loads that are 120V, use a common step-up/step-down transformer.
Cost: est. $250 per building for new bulbs, ballasts, outlets, plugs, power supplies and a step-up/down transformer.
Option 3: Go 240V, install a 120V transformer permanently
This is like Option 2, except with a permanent 120V transformer feeding a third panel.
I often find 5KVA transformers around "used" for about $100. These have 240V on the primary, and 20 amps of 120/240V split-phase on the secondary (jumpable for 40 amps of 120V only). Do the sub-panel per option 2, and add a 20A breaker to feed this transformer. The third panel gets its hots and neutral from the secondary of the transformer. It gets its ground from the ground rods feeding the sub-panel. Because the third panel is fed from a transformer, it's actually not a sub-panel at all - it's a main panel, and you do bond neutral to ground here. The transformer makes this safe.
You only have 20A per leg (or 40A at 120V-only) - so keep the loads here to a minimum.
Cost: est. $300 per building for used transformer, enclosure, additional mini-panel, breakers and wiring.
Option 4: Use a transformer for all loads.
Here, we use a larger transformer (12 KVA or larger, single phase) to feed the entire sub-panel. Like above, this makes it a main panel.
In the house's main panel, we mark the white wire with tape, and punch it down into a 240V breaker. Ground goes to the ground bar. At the transformer, we jumper the transformer's primary for 240V and attach the black and marked-white wire. Ground goes to the transformer chassis or enclosure.
We jumper the transformer's secondary for 120/240 split-phase. Its hots and neutral go to the 2 hots and neutral bar. Being fed by a transformer makes it a main panel, so ground and neutral are bonded together and the ground must go to the rods and also to the transformer chassis. The now-main panel is hooked up in the normal way. This gives you best of all worlds at higher cost.
Cost: est. $500 per building for a used transformer and wiring; $1000 for a new one.
Option 5: Pull 4 wires (in existing conduit??)
ThreePhaseEel addresses this well in his answer. You're pushing the limits of what that conduit can handle, out and in, and it's going to be one miserable pull. I prefer to install conduit much larger than the wires need, mainly to make the pull easier (and provide room for expansion). I would bring in an electrician for this - simply because they'll have all the right pulling tools on the truck, and the skill to do the pull without tearing up the wire. The right tools make all the difference in the world.
Keep in mind if there's any damage to this conduit, the pull will fail and you will end up digging up the yard. Water in the conduit is no big deal; that's why you use THWN wire.
And don't presume it's in conduit. If it's direct-burial cable, Code requires the cable exit the ground in conduit. That would explain using barely-big-enough conduit that he couldn't possibly pull 200 feet through. If it's direct-buried, you'll have to re-trench it.
Cost: est. $700-1200 per building: $400 for wire; $300 for tools or $800 to hire an electrician for the pull. If the conduit is not pullable, more.
Best Answer
You might as well go all the way with your subpanel upgrade
There is absolutely no reason to put a dinky 10-space panel in your shed when you could get a 24-space or 30-space, 100 or 125A, main breaker panel for a minimal additional cost (the price differential we're looking at here pays for oh, a pizza or a few fancy cups of coffee, depending on where you are). Keep in mind that the main breaker in your subpanel is simply a shutoff means for the shed, so it being larger than the feeder breaker in the main panel is not an issue. Don't forget to buy and install the correct grounding bar for your panel if it doesn't come with one pre-installed!
There are better ways to invest in copper
You're far better off pulling a 2-2-4-6 Al mobile home feeder cable for your run than doing anything with copper. It's one-third the price, has 50% more current-carrying capacity (90A at 75°C), and will happily fit in 1¼" Schedule 80 PVC conduit.
While you're at it, you might want to use some of that money you saved by not running copper to lay a continuous run of PVC conduit belowground. While 1¼" will work for this, going up to 1½" gives you room for future upgrades, up to 100 or 125A even depending on the panel you get.
At the main panel, you'll need an 90A breaker of the correct type for your panel (although you can use a 60A breaker if you already have it fitted) to take full advantage of the mobile home feeder cable.
As to that grounding situation...
You are correct that you'll need ground rods at the shed in addition to the grounding wire in your feeder cable. This is handled with some bare 6AWG copper run from the garage subpanel's grounding bar to two 8' ground rods driven 6-8' apart, as a single rod might not get a low enough resistance to earth to protect your shed adequately against wayward natural electricity. Don't forget to pull the green bonding screw or strap out of the subpanel while you're doing this installation, for that matter!
TORQUE ALL LUGS TO SPEC
Last but not least, you'll want to use an inch-pound torque wrench or torque screwdriver to tighten all connection (breaker, panel) lug screws to the rated tightening torques marked on the breaker or panel labels. This is required in 2017 NEC 110.14(D), and is a good idea even if you're on a prior edition of the NEC, lest your electrical work give your garage the loose lugnut!