That sounds like arthritis to me, but he seems a little young for it. Have you have him for five years, or is the 5 year age an estimate given you by a vet?

Cosequin is a joint support medicine/supplement often recommended (no RX needed, IIRC) for animal joint issues. Many people with older, arthritic animals, swear by it. You could give it a shot, but I'd still recommend a vet especially since it's been going on for a couple weeks.

Good luck :(

Cats can also get hip dysplesia. You would need to take him to a vet for a diagnosis.

This sounds a little like my cat of several years ago. His back legs became swollen, and his gait was a little off, and instead of running everywhere he would slowly walk. The legs & hips & base of his tail were hot to touch too. My vet diagnosed a calcium deficiency, not due to dietary insufficiency but due to some rare condition that causes calcium deficiency that was related to him being FIV+ve.

Calcium deficiency is extremely rare, so it's unlikely to be that and unsafe to assume it is, but the presentation is similar. Your cat sounds a lot more unsteady than Basil though. He could sit with no difficulty, and Basil didn't tip to lie down. I would be wondering more about some spinal nerve damage, either due to injury or disease process, especially since it is both legs. I hope you can find a way to take him to the vet.