You originally had an Asko washer/dryer matched set. This wasn't just style. The Asko washer plugged into the Asko dryer, which provided a special NEMA 6-20 receptacle on the back, specifically for an Asko washer. Why? The set is designed for high-density condos/apartments, to minimize utility hookups (no hot water, no dryer vent, no separate washer power). See page 18 of this document.
My guess is, you live in a large housing complex which found it profitable to buy the exotic Asko units to spend less on hookups. I'm also guessing you "own" rather than "rent" since a landlord would be more hands-on.
Was the electrical connection part of your buying decision? I'm sure the appliance store has sold into your housing complex before. Did they know about the peculiar Asko hookup and recommend the LG because it's compatible? If so, it should have a 6-20 outlet on the back; done.
Otherwise, aside from consumer or legal recourse against the appliance store for selling you the wrong thing, let's talk about your technical options.
Obtain a dryer that is compatible with the Asko "plug washer into dryer" arrangement. Your maintenance department may have suggestions.
Have maintenance pull a new wiring run to an additional NEMA 6-20 receptacle for the washer. This will be expensive (remember, this is why the complex spent extra on Asko units) but will give flexibility - letting you choose a wider selection of washer/dryers. (they still need to be water-heating/ventless).
Your dual-outlet solution is illegal and unsafe because the dryer could pull 30A while the washer pulls 15A. The Asko dryer was designed to share a 30A circuit with the washer it controls - the LG isn't. I suppose it might be possible to get a very large switch to power EITHER the 14-30 dryer outlet OR the 6-20 washer outlet. But insurance/liability/HOA won't let you homebrew that, you'd have to hire it done, and it'd cost as much as option 2. Plus it'd be rather "hillbilly".
Don't get adapter cables and unplug the washer and plug in the dryer every load. These large outlets are not made for frequent unplugging.
I would just ditch the dryer and hang clothesline... in the kid's room... the one who broke the dryer!
NEMA 10-30 receptacles are grandfathered, illegal for new installation but legal to leave alone if already installed. It is not safe and it has a body count. It is legal because of lobbying by the appliance industry.
By modifying it you lose your grandfathering and make it illegal.
Your proposal solves the safety problem electrically, but mechanically it is not code legal. I am reluctant to say it would be "safe" because generally, code is written in ash and blood, and they wouldn't outlaw a thing unless experience has shown it causes fire or fatality.
Your best bet is to retrofit a NEMA 14-30 4-pin receptacle, and connect hot-hot-neutral to those same wires on the 14-30. Connect ground as follows:
Code (as of NEC 2014) allows you to retrofit just a ground wire, simply by running a solo ground wire. It does not need to use the same route as the conductors. It cannot simply go to the rod/pipe ground source that the panel uses. It must go back to the panel, and the same panel as the conductors.
The ground path must be the required wire size its entire run. As long as that is true, multiple circuits (out of the same panel) can share a ground wire when retrofitting grounds.
Best Answer
Given the proximity of the 10-30R receptacle to the panel, and the presence of conduit connecting the two, you could upgrade the 10-30R to a 14-30R and likewise swap out the power cord on the dryer. That's something like US$25 in parts. This is an upgrade that is just good sense to implement.
If you upgrade to a 14-30R then you could easily make or buy an adapter from the 14-30P to L6-30R. It would deal with the ground-vs-neutral situation correctly.
There's also another option. Since the panel is just right there, get a bit of conduit and a box and install a new L6-30R receptacle. I believe code allows multiple 30A receptacles to be fed from a 30A breaker. If the panel has no room for a new breaker and you don't want to replace existing breakers with a quad breaker solution, you could choose to feed both receptacles from the one existing breaker. The terminal on your breaker might be designed for landing two conductors, but if not, attach a pigtail of wire to the breaker and use a wire nut to join the pigtail with the leads going to the two receptacles (times two: this same arrangement for both poles of the breaker).
The two-receptacles-one-breaker arrangement could be accomplished by putting the junction in the box with the dryer outlet but it might violate box fill rules and definitely would be crowded and hard to work with. It'll be much easier to run the conductors from a new receptacle directly into the breaker box and make the junction there.