You need to buy a 2 amp trickle charger and put it on the engine battery which is not getting charged while hooked to shore power. Drawing the battery down in 24 yours suggests you have a light on (maybe in the basement) that's killing your battery although a new batter is often not fully charged by the time it's sold.
Engine start batteries are not deep cycle nor 6V. You should have a boost switch on the dash whch I call a automatic jumper cable as it connects your house and engine batteries together for more starting amps. It can take 10 minutes of holding it on before it will charge the engine battery enough to turn the engine over.