Is it true that shaving under your arms makes it grow thicker?

Kinda-sorta, but not really.

The hair that you have has been out and about for awhile. It's gotten chafed against clothes, bedclothes and skin. It's been washed and dried. All these things have worn the hair down a bit, making it softer.

Now, if you were to shave that off, the new hair when it emerges will be brand spanking new, and so a little stiffer and a little coarser.

Give it some time, and it will get worn just like the "old" hair.