Do you still pay 10x the dice when getting the chance card that takes you to the nearest utility if the property is mortgaged?
Yes, you have to pay.  Consider what the card says:


  Advance token to nearest Utility. If unowned, you may buy it from the Bank. If owned, throw dice and pay owner a total ten times the amount thrown.


This has nothing to do with paying rent.

As you mentioned, there is another Chance card that sends you to a railroad:


  Advance token to the nearest Railroad and pay owner twice the rental to which he/she is otherwise entitled. If Railroad is unowned, you may buy it from the Bank.


This card refers to the amount to which the owner is otherwise entitled, namely rent.  If the property is mortgaged, that amount is zero.  The card you're asking about makes no such reference.