The voltages you mention look a bit low. A good battery should supply close to 12v or a little under. When the car is running you should get to 13-14v. I would try measuring the voltage at the alternator when the engine is running to see what the generator output looks like. You may have poor power or ground wires. It may also be worth disconnecting the battery and measuring it directly, without any load. Getting the battery load tested at an auto parts store is probably a good idea - most will do it for free. Personally if I have any battery like issues and it is over 4 years old I will often replace it as that is the most life I usually get from them. After that you might want to look up videos on parasitic draw testing on google to see if something is draining power.