Several answers... depends upon what fix you are trying to identify. Your method works but is not necessary. It is true that the GPS will generally NOT be giving you the distance from the actual localizer DME antenna. Instead it is indicating distance to the next fix on the approach, which is generally what you really want anyway.
So, for example, if you are doing a localizer approach and the LOM is at "7.6 DME" you can just ignore the 7.6 and watch the GPS to indicate waypoint passage at the LOM.
After passing the final approach fix the GPS will NOT indicate any further stepdown fixes. It will count down the mileage to the missed approach point (MAP). That is where you have to read the chart carefully... to identify the stepdown fix you need to know its distance from the MAP. This is generally NOT the same as the "DME" distance, which is distance from the DME antenna. You may have to do a little math, this should be part of your approach brief.