In reference to the battery, it is a good point, but if you have a battery tender or charger charge it up completly and see if any of your issues are corrected before you just replace things chasing a problem, I'd even go a step further and charge it up and have it load tested, you can get that done for free.

Also being batteries can be a cause of a lot of different problems, I'd also double check the ground connections are good and tight on clean surfaces as well. Really there is no need to throw money at something that may not be the problem. Just saying.

Harry