Whether you are using 'city water' hose connection OR on-board pump ALL hot water comes out the top fitting/line on water heater tank.
Sounds like your check valve in HOT outlet that is starting to fail.
They are a cheapo plastic junk that can easily jam up when there is any sudden pressure on them.
Seeing as you lose ALL hot water flow when using washing machine it has to be the check valve (washer solenoid popping open/snapping closed)
Turn off water supply, relieve pressure on plumbing system (open faucet hot/cold) then remove top water line from fitting on tank. Remove fitting it has the check valve inside)
Now......
You can replace with another fitting/check valve
You can just gut the internals on fitting and reuse w/o a check valve (which is only needed when bypassing water heater----winterizing)
You can replace with a brass style check valve
You can use a nipple with a new shutoff valve in hot line
Is it time for your medication or mine?
2007 DODGE 3500 QC SRW 5.9L CTD In-Bed 'quiet gen'
2007 HitchHiker II 32.5 UKTG 2000W Xantex Inverter
US NAVY------USS Decatur DDG31