If the accessory battery is not charging while hooked up to shore power, take a multimeter ad check the voltage coming out of the DC ports checking from + to - (or black to red) it should be around 14vDC. If its less, the converter needs replaced.
pull the spark plug after trying to start it. It should either be dry or have gas on it. If its dry, the carb problably needs cleaned by soaking it in carb cleaner, and then blowing out every passage way with compressed air. If it wet with gas, try a new plug and see if it fires while holding it on the head, while pulling the starter rope. If it fires, replace the plug and try to restart it.