I'd try keep my right foot outside of his and circle to the right. This is good for a number of reasons, by doing so you'll have some good opportunities by staying outside of his left, and you'll be stumbling less when you step to your right.

From there you can take a quick step to the right and hit him with a right hook. Continue with another pivot/step the right to get to his side (this a good position to be in) and if you see an opening throw a hard straight left, or even a left uppercut between his gloves would be good.

Since your opponent brings his hands up to his face, he's not paying much attention to what's coming next. So try a 1-2 to get his hands up, and then step to his side and pop him with a quick right hook to the body or head. I'd also practice following the right hook with a straight left and vice-versa.

Anyways It's not bad to have your right hand down from time to so long you're jabbing, and you're using pivots and head movements to avoid their leads. I'd try to keep it up in the same way that you do when you're about to catch a ball. With it there, you can use it more assertively, both defensively and offensively. You can knock down their jab with your right hand, and then fire the jab over it while you're stepping to the right.

Don't be afraid to jab to the body. The jab to the solar-plexus (that's the sensitive spot below where your ribs meet), gives them sickening feeling that they won't like! Try a quick jab their solar-plexus and then if they bring their hands down, throw straight left. Once they become self-conscientious about the jab to the body, you can feint with your right, which might be just jerking your right shoulder forward and half jabbing, and then if it works hit them with your left again.

I liked how you bent at the waist to avoid their punches, you should watch James Toney's fight with Vasilliy Jirov. He'd lean low to the side under Jirov's southpaw jabs.

Anyways, these are just some of things that you can try.