Battery at almost 12 v sounds like it is shot. Should be at least mid 12 volts. 11.9 is equivalent to 40 %. Is it being charged by the converter as well as solar? Converter output should be 13.x when plugged into shore power to charge the battery. You should measure that at the battery terminals at that time. if you don't have that at the battery, it sounds like the battery is not being charged by your converter and also not enough by your solar.