The airlines are different in how they handle this. You don't say which you normally use so it is difficult to make specific recommendations. For most of the majors, exit rows are either an extra purchase or a perk for their frequent flyers and you will normally not be able to reserve exit seating by phone unless you have the necessary status. That may be your problem. Before I reached the necessary status level on my usual carrier, I had pretty good luck showing up at the airport well in advance of my flight time - several hours - and asking at check-in. Except for those frequent flyers who have the ability to reserve exit seating in advance, most airlines require the check-in agent to see you and verify that you qualify for the exit row - physical ability to handle the door, ability to understand instructions in english, over age 15 and so on.

It is not surprising that you haven't had much luck on the phone. Exits are supposed to be assigned in person. Try asking in person. If a seat is available, you might get it.