Wellbutrin is an antidepressant, and is not usually used to treat ADHD unless stimulant treatment has proven unsuccesfull. It's not the right treatment for someone with a history of ADHD that was successfully managed with stimulants, unless there is also evidence of depression, or there's a good reason that stimulants aren't an option. Basically, I think your doctor is full of sh*t. But you can't very well walk into a clinic you have no existing association with and ask for a prescription for a specific Schedule II controlled substance, particularly when you fit a demographic that's considered particular high risk for abuse, so given that you can't afford to see a different doctor in a private practice, you're probably stuck with the health center doctor. So all you can really do is be honest with him and hope for the best.

It might be worth talking to your doctor about Strattera, which is a non-stimulant treatment for ADHD. Personally I had a bad experience with it (the side-effects were so bad I stopped taking it after four days), and I've heard mixed reports about it's efficacy, but it does work well for some people. But more importantly, it has a low potential for abuse. So if you go in asking for Strattera rather than Adderall, it will be obvious to your doctor that you're not just looking to get high. If you try the Strattera and it doesn't work out, then your doctor will probably be more comfortable giving you the Adderall since you've demonstrated that you're interested in controlling your ADHD symptoms, rather than getting high or improving your ability to pull all-nighters. It's a daily thing like the Wellbutrin, but even if you don't end up liking it, it's still worth trying, and maybe it will turn out to be perfect for you, and you won't want to go back to the Adderall.
References :