1980 Rivvy Starter Motor "Slipping"?

I'm not sure what to call it, but I have a slight problem with my Rivvy.

Info: 1980 Buick Riviera, A/C Removed, Everything stock but stereo. Been working on it since beginning of year, fixed pretty much everything except this and a few other minor issues. It's never had any serious maintenance done to it (oil changes, tire changes etc., I have the entire paper trail).

Anyhow, ever since I purchased this car it's had a starting issue. If it's warm it has no problems. Cold starting is a whole different story though. I normally pump the pedal once or twice, turn the key, the engine turns over 1-3 times then it "slips". Essentially it seems to me that the starter is disengaging from the flywheel and it just spins like crazy while the engine sits idle. I let sit until the starter stops, turn key, and again it turns over a few times then slips. I normally get it on the third time, it'll turn over and then finally start and there's no issues after that.

I was thinking maybe just find a new starter and replace it but I'm not sure how much that's going to run me and if it's a common problem that's fixed reasonably easily I'd rather fix it then put a new starter in :).

Thanks in advance for any help, fyi I do have all the tools required for the job (I think) and a jack and stands and the such.