Nothing wrong with using 'do do" but it just sounds like something doggies do. :-)

The above sentence: I don't do anything I am ashamed of, but sometimes I do do things I find embarrassing.

Might sound better if said this way: I don't do anything I am ashamed of, but sometimes I actually do things I find embarrassing.

Nothing wrong with using 'do do" but it just sounds like something doggies do. :-)

The above sentence:

I don't do anything I am ashamed of, but sometimes I do do things I find embarrassing.

Might sound better if said this way:

I don't do anything I am ashamed of, but sometimes I actually do things I find embarrassing.