"Dry up" would not be used with drying your hands. "Dry your hands with the paper towel is correct." The 'up' part of the phrasal verb would be connected a stated object of what is being dried up. Part of a children's song called the Itsy-Bitsy Spider: "...Out came the sun and dried up all the rain." So "dry up" needs to be connected with water, or whatever else is causing wetness. Dry up: "Use the paper towel to dry up the water on the floor" or Dry: "Use the paper towel to dry the floor."

For the second sentence you can use 'dry' or 'dry out'. 'Dry up' is not correct. 'Dry up' would be used if something lost all its moisture and became shrivelled up. It doesn't make any sense with clothes. We need to hang your t-shirt on the (washing) line so it can dry (out.) I agree 1a is the only one that makes sense by the way. 1b and 1c sound bad, especially 1b.

Yeah, I would say "dry up" is not said. "Dry your hands with the towel." and "We need to hang your T-shirt on a string so it can dry." are correct.

Of your #1s, only #1 is correct, not b or c. If you said those to a native speaker you'd get a very odd look for sure. Regarding #2: We need to hang your t-shirt on the clothesline to dry. Dry up and drying up is another thing altogether. Maybe someone else will address that for you. Udachi

1. "Dry your hands with the towel" 2. "We need to hang your T-shirt on a string so that it can dry" The 'up' is typically used in a past tense scenario. Ex: The lake dried up.