So if the double pull is from the outside handle, you need to check the latch to handle cable adjustment. The exterior handle connects to the latch mechanism with an adjustable cable.
Since you have a 2003 convertible, I have one as well, your drivers door latch has been changed at least once, if not twice.
There are very few people that can properly replace the door latch in these cars and get everything set up and adjuster properly.
Compare the drivers door to the passenger door for how for the door handle is pulled for the latch to release/open the door.
The good news if the cable needs adjustment, you do not need to open the door up. All you have to do is remove 1 small triple square/XZN screw and remove the door lock cylinder, then the you can adjust the door latch cable.