First of all, DON'T buy stuff before you try going to stock rom, you can get back anytime from your nandroid backup. Battery problems are very hard to track, someone already suggested betterbatterystats, try finding it yourself with that, and if you fail you can;
1- Wipe everything and run the phone with only betterbattery stats installed and see the problem,
2- Go to stock rom and see if your problem persists without any apps installed.
Edit: If your cpu is misbehaving (even in stock), there is nothing you can do since its a hardware problem, you need to go to service centre.
Phone: Nexus 5 16GB
Tablet: Nexus 7 32GB
Old Android Phones: Samsung Galaxy Mini, Samsung S2 I9100G