You can't just throw in a bigger breaker, and expect it to work the way you want.
First of all, you're going to have to have the utility install conductors sized appropriately for 200 ampere service. Depending on how the building is wired, you'll also have to upgrade the conductors that run from the service drop to the meter (this might be handled by the utility, depending on how the building is wired). Next, you're going to have to upgrade the conductors that run between the meter and the disconnect (likely handled by installing the new panel pictured). Finally, you may have to upgrade some of your grounding and bonding conductors, as they are typically sized based on the size of the service.
Once all that's done, then you can likely proceed as you've described. However, you're going to have to evaluate how the system is grounded. The service neutral will have to be bonded in the new panel, where the 200 amp disconnect lives. Then you'll have to feed the existing panel (now a secondary panel), with a 4 wire feeder (hot, hot, neutral, ground). You'll also have to isolate the grounded (neutral) bar in the existing panel, if it's not already.
Long story short, this is not a simple job. There are a lot of subtle details, that can easily be missed by a DIYer. I would recommend having an Electrician at least help you plan the project, and inspect the completed work.
This one popped up for some reason, so I'll edit in some new thoughts.
Corroded or arcing power feeds are "burn your house down" serious business. Fix immediately! Aluminum feeder wire is fine stuff, but you do need to use the anti-ox goop, and torque it correctly. Many electricians in the 80s and 90s did not get the memo. Nip it in the bud fast.
Likewise corrosion (read: arcing) on breaker contact points is absolutely unacceptable. Catch it early and you may be able to save the breaker space, otherwise that's a dead space in your panel. The #1 cause is using the wrong brand of breaker. Brand X breaker won't engage Brand Y buses with the correct contact shape or clamping force, causing the arcing.
If the panel is a goner, swap it. Let me come back to that.
The cable between main and sub doesn't need 2 breakers. It needs a breaker upstream (in the main panel) to protect the cable. A 60A breaker is correct to protect #6Cu or #4Al. At the subpanel, the cable/wire can land on main lugs (provided the subpanel is rated for 60A+) or on the subpanel's main breaker (of any rating - a smaller main protects the panel, a larger main is just a shutoff switch, which you do not need since it's in the same building.) So if the subpanel has main lugs, use them instead of backfeeding a breaker.
One more thing. If the main panel is breakered at 100A, and the cable supports 100A, and the subpanel supports 100A, then the main main breaker protects them all. You don't need any breakers for the sub. You can use thru-lugs or snap-in lugs in the main panel.
With that in mind, back to replacing the panel(s). Talk to the power company and see if your service drop/meter can support 125A. Even if not, you can order a 125A main panel and swap the main breaker to 100A.
See if you can upgrade the main-sub cable to 125A. If so, order a main panel with thru lugs, which will save 2 breaker spaces, and a 125A sub. Voila, no breakers needed, space maximized!
You can also look at the CH or QO panels, which use 3/4" high breakers, so more spaces in the same panel footprint. GFCI and AFCI breakers are available for them.
What doesn't work is "double-stuff" breakers (including GE's 1/2" Q-line breakers). Those are useless as almost all new work today requires GFCI, AFCI or both - and you cannot obtain those in double-stuff. So a "24/48" panel is actually 24, period.
Original:
Wire gauge is decided by what NEC requires (and those are in stair-steps) and what your electrical distributor has in stock.
So with numbers as close as 100A vs 125A, you might get lucky: imagine the original installer found #2 wire was good for 95A, #1 for 110A and #0 aka 1/0 was good for 125A. He can't use #2, the distributor does not stock #1, so he uses 1/0. You get a happy surprise when you aim to upgrade. Hey, it could happen - check. If not, you need to pull bigger wire.
Check the markings on the aluminum conductors to make sure they are stranded and the modern AA-8000 series alloy, which are legal and safe. (NEC 310.106b). The problem alloys from the 1950s are now outlawed, and your distributor won't even sell them.
Your reduced voltage to 108 is more likely to be a problem with your neutral. Poor connections can't drop 12 volts without destroying themselves from heat (volts x amps = watts of heat), more likely your other leg went to 132 volts because of a lost neutral. This is an extremely dangerous condition that needs to be taken out of service immediately because the imbalance can be much worse than 12V, and that will destroy appliances and start fires. If you know where the problem is, but can't de-energize your house now, a short-term work-around is disable one "hot" - doesn't matter which one. Look at how the hot busbars are arranged inside the panel, pick one leg, and remove every breaker which uses that leg. Rearrange as necessary to keep the lights on. At that point, if the neutral fails, the circuits will shut off instead of giving dangerous voltages.
Look for other options (i.e. other brands or suppliers) that will give you more spaces in the panel. You don't want to be forced into using duplex breakers, because - especially when you remodel - you'll be forced to follow current code, and will need more kitchen breakers and GFCI and AFCI breakers in many slots. Those are a lot more expensive as duplex. You can completely fill the panel with double-stuff breakers (in fact, doing so is the basis of their "48 circuit" marketing claim).
Don't buy this at the big-box home improvement stores, go to a real electrical distributor such as Greybar - they tend to be locally owned. They will have better options (eg more breaker spaces in smaller boxes, Homelites are huge), better quality and sometimes better prices. For instance, Homelite is Square D's bottom-tier brand. There's better out there, and it's worth an extra $100 to not have problems like the ones you are having.
Best Answer
Call the power company NOW and report a power outage
The fact that you've worked on your panel is unrelated. Given the testing you just did and the results you got -- you have a power outage at the meter pan or before. It could be where the neutral service wire connects to your neutral bar, but that's the only place it could be on your side, and super easy to check.
To be more precise, the type of outage is a "Lost Neutral".
Loss of a supply wire at or before the meter pan is a power outage.
If the dead supply wire is a hot, half your panel will work. If the dead supply wire is a neutral, your 240V loads will work and your 120V loads will have scary weird voltages on them.
It's still a power outage and should be treated as such.
As long as you are confident in the quality of the connection between your neutral service wire and your neutral bus, you should button up the panel and there's no reason to show it to the power company. Everything they'll care about will be at the meter pan or their side of it.
DON'T procrastinate this or try to work around it by balancing loads etc.
Seriously, unless you plan to shimmy up a pole with a set of Allen sockets, every possible path ends with you calling the power company and them fixing their service-drop issue. Which they will do for free, same-day and probably 10 minutes up on the pole. And then you'll be like "darn it Harper, why didn't you lead with 'Call the power company', why send me in circles doing all that other stuff first?"