Whens the last time you switched FOB batteries? If they are getting low they can cause all sort of havoc with your electrical system. If that is not an issue, then I'd next look at Battery cable connections at both ends, just not at the battery. Make sure they are tight and clean. Lastly, get you battery load tested to make sure it's good. If you have the original battery it's 4 years old an may be going out. Weak, dying batteries also cause strange electrical issues.