Not precisely clear what you mean by "constant change". A two-year old won't be overly concerned about the ramifications of the father "constantly giving up his weekends." Certainly, when the child gets older, he'll start to wonder.
"So I would like to file for full custody but don't know if it would just be a waste of time."
It's unclear what you mean by "full custody" but the likelihood of the father having visitation yanked altogether (which is presumably what you're getting at) vs. every other weekend is virtually nil. Naturally, you're free to file, but I do believe it would be a waste of time.
Even if the rehab was court-ordered (you don't say), rehab isn't a bad thing (the history of being in and out of jail simply means that you didn't really get to know him very well, or didn't care about his behavior when you entered into a romantic relationship with him; water under the bridge).