Solar charger but still dead batteries

Try disconnecting the battery ground cable the next time it will sit for a week. If it starts ok when you reconnect the cable, the battery is good and there's a draw somewhere else that's pulling it down. The quick fix in that case would be a battery disconnect switch that installs on the battery terminal, but finding the draw would be the better answer. That can be a long process though, unless you have a good multi-meter on hand and know how to use it for load testing.