Why There Are 240V Outlets in All Rooms of a House

If you're talking about the equipment grounding conductor, and not the grounded (neutral) conductor, then they may not have to "share" the same equipment grounding conductor.

This answer contains all the code references, if you want to give it a read. If your area has adopted the 2014 version of the National Electrical Code, you can share an equipment grounding conductor from another branch circuit, as long as both circuits originate from the same enclosure.

250.130(C) Nongrounding Receptacle Replacement or Branch Circuit Extensions. The equipment grounding conductor of a grounding-type receptacle or a branch-circuit extension shall be permitted to be connected to any of the following:

(4) To an equipment grounding conductor that is part of another branch circuit that originates from the enclosure where the branch circuit for the receptacle or branch circuit originates

First step: receptacle replacement

Your existing receptacle is a 10-30R. This will need to be replaced with a 14-30R in order to plug either of your new appliances in. Furthermore, it appears you have aluminum wiring, so this will be a bit trickier than usual, but not impossible for a DIY job. Of course, if you are more comfortable calling in an electrician to deal with this, they'll be able to take care of it without too much trouble, and may be able to bring better stuff to bear on this job than you can.

For doing this yourself, you'll want:

• A NEMA 14-30R receptacle of the flushmount type
• Some short lengths of 10AWG copper THHN (black/red/white/bare if you want to insist on color codes, or you can use white as a universal donor with some tape flags for hots and wire stripping for the bare ground) for pigtailing the receptacle.
• 3 port Al7Cu or Al9Cu insulated screw lugs rated for 8AWG/10AWG wire
• Anti-oxidant compound if called for by the connectors used
• And a torque screwdriver, as the requirement for it got added in the 2017 NEC (you can get one at the orange borg or online for $60 these days)

Once you have the breaker off, you can undo and pull out the existing receptacle, and also undo the ground from the box lug at this point. Toss the old receptacle as you won't need any more of it. Fit the cut and stripped pigtails to the new receptacle, using your torque screwdriver to torque things down to specification. If called for in the connector instructions, apply anti-oxidant compound to the ends of the existing wires. Then, use the screw lugs to connect the pigtails to the existing wires, matching the pigtails up with the corresponding wires and using the torque screwdriver to torque the terminal screws down to specification. Stuff everything into the box (it may be tight) and screw the new receptacle in before turning the power back on and using a meter to check your work for the correct voltages.

Now: time for two 30A receptacles on one 30A circuit

There is absolutely nothing in the NEC that prohibits placing two 30A receptacles on the same 30A branch circuit. Neither 210.21(B)(3) nor 210.23(B) forbid this -- in fact, you could say that the Code explicitly provides for this exact case (otherwise there wouldn't be a line for 30A receptacles in Table 210.21(B)(3)).

Unfortunately, I don't know enough about how your house is built or where you want the 2nd receptacle on your wall to be able to instruct you further here. (This job will likely require you to tear out a bit of drywall, too.) The instructions above will leave you in a good position to do it or have an electrician do it, though.

Bonus! Your breaker box should be replaced at the earliest opportunity available!

You have a Zinsco panel -- while they don't quite have the sordid failure history of FPE's line of non-breakers, Zinscos should be replaced at the earliest opportunity available as their track record is poor, relative to something like a CH or a QO.