You are mistaken - it is true. The experiment has been done (with airplanes instead of spaceships) and the effect is very small but measurable. The principle is further tested with the clocks on the GPS satellites every day - if we did not allow for time dilation effects the GPS system would not work.
You might also want to take a look at the sticky thread at the top of this subforum:
https://www.physicsforums.com/threads/faq-experimental-basis-of-special-relativity.22903
And this is one of the reasons that I suggested you start with relativity of simultaneity. Google for "Einstein train simultaneity" and see what you find.
You cannot become younger than you were - that would be time going backwards and that's not possible. You can, however, age less quickly than someone else if less time passes on your path through spacetime than the other person's. But as I said, this is a different phenomenon than time dilation - you have to understand the basics first.