Can I switch my washer between two power sources? Double throw switch?
Move the double-pole switch to the washer.

A double-pole switch is just right.   You will switch hot and neutral.  You've already got one for the dryer; move it to the washer. 

Switching neutral is essential when the outlets are fed from different circuits.  And these are served from different services, making it far more critical. 

Get a 3-pole switch for the dryer.

The huge problem with electric dryers is that third wire is not ground. It is neutral.  It is an active conductor that handles normal current for all the 120V loads in the dryer - basically every internal device except the heater.  You must switch neutral.  You have connected the two service panel's neutrals back to back, which means neutral from either panel has two choices of path -- back through the service entrance, or via the dryer connection to the other tenant's panel.  This can cause all manner of problems, and is certainly the most likely way I can think of to set an apartment on fire.  

(or you've connected neutral to only one service, but that has a whole 'nother way to set the house on fire). 



Neutrals do not have circuit breakers.  The safety of the neutral wire depends absolutely on the concept that it only returns current from its circuit.  As soon as you criss-cross neutrals between circuits, you can overload that neutral. There is a variety of ways that could fail deadly.  Including some that could electrify the chassis of the dryer, because of that obsolete, unsafe 3-prong connection! 

Things get even more serious when criss-crossing neutrals on two services, because now the criss-crossing neutral wire could carry all the current for the entire service. That'll start a fire for sure. 

It is all right to criss-cross safety ground wires; in fact if you use metal conduit for your wiring method, it's inevitable. I have a building with 4 separate services all sharing the same metal pipes on a metal building, and thus, same safety ground. But the important thing is that a 3-wire dryer connection has no ground; that wire is neutral. 

Lastly, I very strongly urge you to convert to a 4-wire connection to the dryer, including removing the (technically legal) neutral-ground bootleg strap. Because this type of switching adds a risk of a problem with the neutral (gee, you think? :) and on a 3-wire connection, that causes the chassis of the dryer to become electrified!

Rule of thumb: Don't do anything a GFCI would have a problem with

If you use a wiring technique that would trip a GFCI breaker, particularly sharing neutral, then don't do that. It was already against Code, but the GFCI keeps you honest :)

NEC 2014 already requires GFCI in the laundry room for 120V outlets e.g. washer.  I advise using GFCI protection before the transfer switch, so that everything associated with the switch is protected by GFCI.  That means a GFCI breaker, or a GFCI recep that feeds the transfer switch from its LOAD terminals. 

NEC 2020 will require GFCI for the dryer as well, and the safest (and most affordable, actually) way to do that is before the transfer switch, so again, the transfer switch is in the zone of protection.  That way any screw-ups with the transfer switch are GFCI protected. 

I know 90% of this  has been a "lecture", but I gotta give you more lecture. 

In fact, actually do fit GFCIs... for liability.

Here is where GFCI can save your bacon.  Make sure that both transfer switches are served by GFCIs upstream of the transfer switch e.g. at the breaker - a GFCI switched by the switch is of no help.  However, for that to work, you MUST also convert the dryer to a 4-wire connection, and remove the neutral-ground jumper in the dryer, and also remove any other bootlegged grounds in this area (because those break GFCI protection).  If GFCIs are protecting, you're far better off with no ground than a bootlegged ground.  Neutral and ground must be totally separate for the GFCIs to protect.  Once the GFCIs are installed, it's less important that things actually be grounded, although retrofitting ground would be "belt and suspenders" and I would do it for liability reasons. 

At this point, you've done your part to prevent an electrical shock, and one won't happen.  And if one does anyway:  the investigators will make a  "landlord made full and fair effort, no negligence here, don't charge with manslaughter" recommendation to the DA.  Now you're down to civil fines for violating the procedural stuff about permits and electricians. 

All work in a rental unit must be done by a licensed electrician. That is state law in pretty much every state, and it's to prevent landlords doing their own "hack jobs". You can only do your own work in owner-occupied units. 

Any work of this complexity requires a permit - precisely so the town inspector can keep an eye on the setup to make sure it is done correctly. 

So you've already crossed the line on those; your only defense is doing the work well.