1. Shore power will keep the battery from discharging (trickle charge), but is not meant to recharge a low or dead battery. You need to fully charge them with a proper charger, then they should stay topped off when on 120v
2. I would either contact the dealer and have them check/replace or simply take them to a battery shop and have them test. These may be bad batteries from the dealer and you would need to work this out with them.
3. Batteries have a point (usually 50%) where if you drain them below that minimum charge, you dramatically lower the lifespan (and will eventually need to replace them if continued).
4. Depending on how discharged they are, it may take 24hrs (or more) to fully charge your batteries.
I would charge the batteries individually (not as connected to the trailer). Again, this is using a proper battery charger.
I use a NOCO Genius:
http://www.amazon.com/NOCO-G750-UltraSafe-Battery-Charger/dp/B004LX3AS6/ref=sr_1_12?s=automotive&ie=UTF8&qid=1456341536&sr=1-12&keywords=battery+tender
(but you can use pretty much any brand you like).
5. If your batteries are connected to your trailer, there will always be a "phantom" drain from the CO2 detector etc. but should not drain your "fully" charged batteries in a couple of days if all other things are off. Also, one bad or low battery may drain the other when connected in series or parallel.
Im sure others will chime in and correct anything I have misrepresented......
Happy Camping!