All sorts of things will go wrong.
For one thing, you cannot manually add files to a Time Machine backup.
For another, if you just drag & drop what you remember, you'll forget things, and all sorts of oher things will be missed.
First, is Time Machine deleting old backups, or just expired ones? How old is your oldest backup, and how much space is available on the TM drive?
Second, assuming it really is near full, you need to figure out why. Take a look at #D4 per my previous post -- that lists some commont causes, and a way to see what's getting backed-up repeatedly.
A 500 GB drive is a bit small to back up 300 GB, but depending on how you use your Mac, it should allow several weeks to many months of backups.