engine battery dies

What you have described is pretty common. First, there are some parasitic drains on most engine batteries when the rig is parked. Second, your engine battery is probably NOT charged when the rig is plugged into shore power. The chassis batteries are charged when plugged in but probably not the engine batteries. This is common on many motor homes and manufactured that way. The MH builder will most ofter use a small solar panel on the roof to keep the engine batteries charged but most don't offer much recharge power and die after a few years. There are several ways to remedy the lack of charge to the engine batteries while plugged in. One is install a separate battery charger for these batteries such as the BATTERY MINDER, BATTERY TENDER ETC. These are small chargers designed to maintain the battery charge level, not necessarily to charge a depleted battery. Another remedy is to install an electronic device that allows both chassis and engine batteries to be charged while plugged in. I think that relay is know as a BIRD relay. So---do some research and you will solve this common problem.