Well, clay would be the wrong product to use to remove oxidation.
Oxidation is basically a dead microlayer of paint. Depending on the severity, there may or may not be fresh paint underneath the dead layer. But you will need to expose that fresh layer in order to make the paint look good again.
There are plenty of polishes and compounds available at auto parts store. If you want to try to do this by hand, use a basic "pre wax cleaner" product first. Meguiars and Mothers both offer nice cleaners. Cost is about $7.00. Meguiars comes in a maroon-colored bottle and Mothers is bright red. Use a new terrycloth applicator and don't be afraid to put some elbow grease into it. Try it on a small, inconspicuous area first to make sure things work OK.
There are plenty of more aggressive compounds on the market if you need to kick it up a notch. But beware that many are formulated for machine use and might be too aggressive for hand use. Read the instructions carefully before bustin' out the gold card.
A final option would be to consult a professional detailer to see what they suggest. Many times a detailer can machine polish your paint with great results. It might be worth kickin' down $200 or so to have your car buffed out and detailed. From there all you have to do is maintain the shine.
Hope all this helps. Happy polishing!!