Tango had all of his vaccinations when I rescued him at 7 months. But with his yearly shots, the Vet doesn't give them all at once. He's so small (3.9 lbs) that she prefers to space them out. She gives them in two batches, a month apart.
Jazz is still getting her puppy shots. She's a rescue too. I was told she was on her third round of puppy shots when I rescued her at 11 weeks, but there was no paperwork. My vet took that with a grain of salt! And has spaced her shots over the last month. She'll go in for her last round in two more weeks, when she'll be 18 weeks old.
I just reread your post, and I don't think any of what I said was particularly responsive to your concerns........sorry!