One solution to the tank valve is one that twists on to the existing outlet, like the Valterra #T58 in the photo. Could be just temporary until you get the original valve issue fixed. The original valve seal could need replacing or it could have something blocking it and preventing it from fully closing or it may need some lube. If you don't have an enclosed underbelly the valve is easy to get to.
Air admittance valves (AAV) are notorious for failing and is usually the first suspect. It does sound tho. like the vent pipe has dropped or is plugged. P-traps never need refilling after a dump. If the vent pipe is plugged, air will get drawn through the sink (or shower) drain and empty the traps of water during a tank dumping. I'd suggest getting a quality commercial grade AAV like an Oatey #39017 as in the photo - $15 on ebay or $30 at Lowes. We had an AAV fail on our new TT at under one season. Got a replacement and that failed soon afterwards so then got the good Oatey one and has been fine ever since.
Sounds like your tanks could do with a cleaning with an RV tank cleaning product or maybe the Geo method. Liquid Drano will work great and is safe on plastic and metal and is what our owner's manual says to use. Normally you would drive around to let the tanks slosh around so not sure how effective any cleaning product will be if stationary but should be better than doing nothing.