I wasn't addicted to it, but wanted to stop it because it didn't give me the
feeling of having had a full-night's sleep.

Try using an antihistamine (Xyzal-by prescription only, though) is a good
one. Then, alternate by using an aspirin the second night, etc. That
helped me. Occasionally, it's still necessary to take an antihistamine or
an aspirin, but they are not addictive. (That is, if you're not allergic to
aspirin.)