How do I attach both my washer and dryer to the same NEMA 14-30 receptacle?
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!