If your only source is the movies, one of the annoying gripes about them is they have this deceptive editing that makes it seem like Frodo left the Shire within a relatively short timespan after the birthday party.
Frodo got the ring on the 22nd of September of 3001. He leaves Bag End on the 23rd of September...of 3018.
Interesting, though that still doesn't explain the aging part though. I get that Hobbits age slower than humans but if their life expectancy is only 100 years, the difference should be noticeable right?
Sam aged, Frodo did not. I recall several mentions throughout the books that Frodo was considered unusually young-looking for a 50-year old hobbit, just as Bilbo himself was very young-looking for 111. Owning the ring seems to basically just pause your aging, even if you're not using it.
Sam was also twelve years younger than Frodo, but since Frodo stopped aging for 17 years, he wound up effectively older, physically speaking.