Tortilla press geometry problems
With no pictures of your press it is hard to make suggestions.
But assuming your press has metal plates with loops and a pin for the hinge, and if you can take it apart, you might be able to lengthen the hole of the plates (by lengthen I mean to wollar out but only to an oval instead of a circle in the correct direction, so there is play up and down between the plates but not forward and backward between the plates).
What you are needing is a small (about 1/8 inch) gap between the plates when pressed.

Remove the hinge pin
Take apart the plates
Lengthen the hinge holes on one plate (top plate) - you should be able to do that with a drill and the proper sized bit, or with a metal file
Place plates back together
Insert the pin

Now there will be more space and the plates should be able to close with a small gap for the tortilla.