Sorry. You're going to need to get the kneepads, move the gas dryer, do whatever you've got to do to access it. At that point you have several options.
And if you need to get away from that dryer socket location, you can use surface conduit such as Legrand Wiremold to relocate the socket(s) to somewhere more accessible. No stapling flexible cords to walls.
Also, when coming off the 10AWG wire into a common receptacle, the 10AWG wire probably will not fit on the screws. If so, pigtail with a short length of 12AWG wire. Don't use the backstabs (ever).
Also, since this is a laundry room, you must provide GFCI protection somehow - either at the breaker (expensive, very expensive if a 2-pole breaker); inline with deadfaces (which won't work with 2-pole), or at the first 120V receptacle with GFCI+receptacle combo devices (livefaces).
If you switch to a 120V breaker, you'll end up with a hole in your panel. Fill that with another breaker, they cost $4.
120V w/ ground
Some of the older 3-wire cable has the "neutral" as bare wires. If so, this is legal to use for ground, as follows: Reassign it to ground, reassign the white insulated conductor to neutral, and black to hot. Install a 20A/120V breaker in the panel. Land the bare on the ground bar (this may be the same bar as the neutral bar), white (neutral) to the neutral bar or GFCI breaker, and black to the breaker.
Retrofit a ground
Really, this is the "gold standard" way to do this thing. It is now legal to retrofit just a ground wire so you have a proper ground. Now you change that NEMA 10-30 receptacle to a NEMA 14-30 receptacle with neutral and ground separate.
Now for overcurrent protection, you have two choices.
- install a PDU (Power Distribution Unit) intended to support a number of servers on a rack. This is a legal way to split a 30A/240V grounded service into multiple 120V/15A.
- change the breaker to 15A or 20A, and extend with the splitter cable you're thinking of. Keep in mind if you use a 20A breaker, you must extend with 12AWG wire, not 14, and on each side you must have 0 sockets, 2+ sockets, or a single 20A (T-shaped neutral) socket.
Bring the existing cable plus a retrofitted ground out to a new subpanel. From there you branch off whatever circuits you want. In this case, the subpanel is providing overcurrent protection and any GFCI/AFCI that may be required by code.
Ungrounded MWBC plus GFCI protection
I doubt the legality of this, but you could hypothetically branch the neutral with each hot, providing two ungrounded sub-circuits of a multi-wire branch circuit. Then bring that into a GFCI device (e.g. a GFCI+receptacle combo device) and label it "No Equipment Ground". This is certainly not legal for new work, but safetywise, the GFCI would do much to make up for the lack of ground. You would need to change the breaker to 15A or 20A as discussed above, continuing with the 30A breaker is right out.
Cheater cable as you propose
It's not legal, but since I know you're going to consider it anyway, to render that safer, change the breaker to 15A. Use a 2-pole breaker, since this thing will be essentially a "multi-wire branch circuit", and those need common maintenance shut-off. Do not use two 1-pole's.
I am also not going to tell you to fabricate a pendant (cheater cord) using a NEMA 10 cord, square 4" deep or 4-11/16" box and a riser cover with a GFCI+receptacle combo (or two of them, one each leg) mounted in it... because that's not legal either.