Consider the sleeves of your jacket to be permanently contaminated. (Though hopefully the inside of your elbow is contaminated only by your own coughing.) Repeatedly contaminating the sleeves doesn't make them worse; just try to avoid having them touch anything else important. (Theoretically you could be worsening the doorknobs, but it seems unlikely you're adding anything they don't already have.) It may not be a perfect solution, but I think it's pretty good.
Regarding washing hands, I've found that if I do that too much then my skin (starting on the back of my hands) tends to dry out and flake off. Certain kinds of moisturizing soap are less bad at this, but I don't choose what soap is available except at home, and it has led me to find other solutions like the above.
Avoid laurel sulfate based soaps. C12 is particularly well sized to make micelles that disrupt the skin. Castille soap/ stuff made from good ol' fats and lye are the most gentle, laureth/pegylated sulfates are less-bad than lauryl sulfates.
Regarding washing hands, I've found that if I do that too much then my skin (starting on the back of my hands) tends to dry out and flake off. Certain kinds of moisturizing soap are less bad at this, but I don't choose what soap is available except at home, and it has led me to find other solutions like the above.