Time Machine fills backup disk but does not complete

I was having an issue with my old HDD connected to my iMac for Time Machine backups. I've since replaced it with a new SSD and having the same issue - also using a new Thunderbolt cable. It is a brand new 1TB drive. It completes a first backup fine. Then it continues to backup endlessly. After a few hours it has now filled the whole 1TB drive. Yet there only seems to be one backup available when I go into 'Browse Time Machine Backups' and that is the first - all the others are greyed out and inaccessible. Now it will no longer backup as it says the disk is full - but that seems to be with repeat, useless, inaccessible backups.