time machine backups missing

imac 2009, 21.5 inch, OSX Yosemite 10.10.5

I'm replacing my computer soon, after many good years, and one thing that's always giving me peace of mind is my Time Machine backups on an external drive (Seagate Free Agent 1TB). My computer had about 800GB on it, and I was getting messages that the disk was full and could not do a full backup. So I removed 200GB+ from computer, deleted many backups that I didn't need, and changed my settings to not back up things like Dropbox, another external drive, and more. I got the backup size down close to 650GB.

But now, it's still not backing up the full thing AND more troubling, my most recent backup is Jan. 28. So after years of diligently backing up my computer, I see no previous versions. As far as I know, I don't need anything from the past that I don't have now, but what can I do to see what backups exist? "Get info" says 760GB is being used, and yet when I open the application, it shows only two backups. Any ideas?

Thanks.