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!
Since your 100 amp, 240 volt service consists of 100 amp, 120 volt branches, you can have 100 amps of load on each of the two branches. Note that the AC has two 120 volt breakers with the handles connected together. That uses both of the 120 volt branches to make a 30 amp, 240 volt breaker. As shown the box has breakers totaling 75 amps on one branch and 80 amps on the other. If you add a 60 amp, 240 volt breaker you will have 135 amps on one side and 140 amps on the other.
You can have breaker capacity in excess of the main circuit breaker rating based on the assumption that many of the load devices will only be used for short intervals of time during a day. However if the server uses 48 amps all of the time, it would not be difficult to imagine a day when the AC is running a lot, the refrigerator is cycling on occasionally, you stick something in the microwave and your wife is vacuuming.
It may not be out of the question, but it is somewhat doubtful. The service is already nut up to today's code. Today, a residence should have two 20 amp circuits dedicated to kitchen counter outlets and one 20 amp circuit dedicated to bathroom outlets. A microwave and garbage disposer often have their own circuits.
The box itself is likely capable of handling 200 amps at 240 volts, but you may have service entry wiring only rated for 100 amps. An electrician may come and quote a price at no charge. You can not get a good answer without that, but there is likely a way to get it done. Note that 48 amps at 240 volts is a lot of power. That is like having the oven and all the burners going at one time in an electric stove.
You don't need the Hubbell 363R6W. Use a standard electric range outlet.
Best Answer
Actually, you don't need to change anything. All 120V 20 Amp receptacles are designed to accept 120V 15 Amp and 120V 20 Amp plugs. Because of that, all 120V 15 Amp devices are normally designed to work properly in a 120V 15 Amp or 120V 20 Amp circuit. This is true for appliances like a washing machine but even for something as small as a cell phone charger that actually uses < 1 Amp.
Once you get past 20 Amp, the design of the sockets change so that you can't mix and match. And you can't plug a 20 Amp device into a 15 Amp receptacle - it won't fit.
Bottom line: No problem.