Fastest mile keeps changing

Probably you never accepted any better personal record than that. You can do it for example on the page Personal Records of Garmin Connect Web.

I do though. I accept it every time it says new record. So can can my fastest mile record keep getting slower?

Do you accept it on the watch or in Garmin Connect?

if first, you need to do it either in the web or in the app. Setting the green mark on the watch you will only wir the pop up message. 
if you do it online (Web/App) the watch forum is the wrong place for your problem. 

It comes up on the watch saying new record fastest mile. Then I accept it on the app

Not only you have to accept it from the app, but you have to be sure that the Web application is set to display all records. On the Web, log in to Garmin Connect; in the left navigation panel, click on personal records, then click on customize list, and make sure all records type are selected.

I had this issue where I had unselected the longest run record, and my watch kept giving me new longest run records (as my runs were apparently longer than some past run). I kept accepting the record, but it was not sticking. Not only was I asked to re-accept the record everytime I checked the activity, but I kept receiving new records that sometimes were shorter that the previous one.

I fixed the selection on the Web app, and now the problem has disappeared. Hopefully this will work for you.,

I don't understand why this would be a thing, I never had this issue with my forerunner 945, but with my 265 every time I run a mile it tells me it's my fastest mile despite me having much faster records recorded on my previous watch the 945.  Garmin should fix this, I am fine with my new watch keep records from that watch, but I shouldn't have to go in and accept every record in the app.  Fix this Garmin!

Sending the records to your new watch, as advised above, should fix it.