The dissolving of ammonium nitrate is often used in an instant cold pack used to relieve swelling.
if this process is an endothermic reaction, i. e. absorbs heat, then why does it feel cold to the touch instead of hot?

Here's what I get  

Explanation:

Does an ice cube feel cold to the hand?

The melting of ice is an endothermic process.

If you hold an ice cube, heat flows from your hand to the ice. When your hand loses heat, it feels cold.

In the same way, the endothermic dissolving of ammonium nitrate removes the heat from your hand, so the pack feels cold.