When immersing on Shabbos, is it permitted to wear ones undershirt or wrap a towel around himself while still partially in the Mikva, in a way that they will get wet? May one put on his clothing with wet hands?

Rema (O.H. 301: 46) rules that one may not walk close to a place where one may slip ans fall in the water and come to squeeze out the water from his clothing. Poskim indeed warn of walking close to a slippery are next to a swimming pool or a lake for that reason. (See 39 Melochos p. 724 and others).