The battery could be getting old. It's not only a matter of time, it's a matter of treatment. If you always use it until the phone tells you to recharge it (10% or so charge left), you're killing it. You should recharge the battery when it gets down to no less than 40%. (See
Battery University - How to Prolong Lithium-based Batteries - the second chart - and do the arithmetic.)
Why is the phone slowing down? If could be a hardware problem, it could be system cache, it could be insufficient RAM (1GB is getting tight for running a few of today's very sloppily-written [IOW larger than they should be] apps), it could be the individual apps' caches, it could be that you're running out of free storage space (the phone needs a certain amount free to operate efficiently) ... diagnosing "slowness" isn't easy, and it has to be done with the phone in hand.
As far as backing up, plug the phone into a computer. (Install
Motorola - Mac & Windows Drivers in the computer.) Copy files (pictures, documents, etc.) to the computer - just drag & drop. Back up your apps and data with
Helium. (If the phone isn't rooted, Helium will give you a link to download a Windows program to allow it to work.) Copy the carbon folder (that's what Helium used to be called) to the computer. Use
SMS Backup & Restore to back up your texts. Copy its folder to the computer. Your phone numbers should all be Google account entries, so they'll be on
Google Contacts, and they'll sync to the phone after you reset it. (If you have contacts that aren't Google contacts [IOW, Phone or Device contacts], use
MyPhoneExplorer Client. Sync to that. Then you can sync back to the phone after you reset it or erase anything. (You can also change the type of contact with that app, on the PC side - it's the only way I've found to do that. You copy, say, a Device contact, paste it back and change the type of the pasted contact to Google. Then delete the original. Why Google chose to not let us edit the type of contact is something I'm not going to try to figure out. After writing software for 40years, I'm crazy enough already.)