Question:   If a beaker contains water and a solute that dissolves endothermically in water, would the beaker feel hot or cold?

Answer:

It's cold.  Endothermic reactions absorb heat from their surroundings.  As a result, the amount of heat in the reaction increases, having pulled energy away from the things nearby.  If your hand is nearby, the transfer of energy from your hand to the reaction will cause yorur hand to become cold.  The fact that DH is positive for endothermic reactions reflects the fact that the reaction absorbs energy from its surroundings.

Likewise, exothermic reactions feel hot because heat is released by them into their surroundings.  DH is negative for exothermic reactions because the reaction loses energy - this energy is transferred into whatever is nearby.