I have a charger, "this accessory is not supported"

Actually, this happened to me too. Several years ago I bought a Mophie brick. It worked wonderfully for about a year. Then, suddenly, I think soon after one of the OS updates, I started getting the This device is not supported message also. My iPhone 5 was also continually running low on juice faster and faster around that time until I recently had to have the battery replaced.

This usually happens with 3rd party changers, make sure somewhere on the box it shows "works with iPod/iPhone"

Is this an Apple charger or a 3rd party.

By "charger" I assume you mean "Cable." It is NOT a charger. The charger is built into the phone. The device that plugs into the wall is not a charger either; it is a USB power adapter. For the moment I will assume you are talking about the cable.

Either it is NOT an approved cable, or the wall adapter is not an approved adapter, or you have a problem with the Lightning connector on the phone. The most common cause of that error message (assuming the cable and adapter are approved devices) is either dirt in the connector that is preventing the plug on the cable from seating properly, or a damaged pin in the connector. So the first step is to clean the connector. Use a bright light and a toothpick, or an air blast can. Inspect afterwards to make sure there is no cruft left in the connector.

I was never able to fix it. And I became afraid there was something in the new OS that would damage the phone if I continued to use it so I stopped. Any thoughts?

Thoughts about what?

3rd party products not approved by Apple may or may not meet the usage and safety specifications that Apple demands of the 3rd party manufacturers. Use them at your own risk, but don't be surprised if they cause damage to your phone and Apple doesn't honor your warranty.