The problem is often not the wrap itself. Tomatoes, cucumber, warm vegetables, and a thin dressing can supply enough moisture to soften even a sturdy tortilla.
Dry washed leaves, drain beans well, and let cooked fillings cool appropriately before packing. Keep very wet ingredients in a separate container. Use a spread thick enough to stay put and avoid filling the wrap beyond what it can hold.
For an immediate meal, you can be more relaxed. A warm, juicy wrap eaten over a plate is different from one expected to survive several hours in a lunch bag.
Chilling remains necessary for perishable fillings. A neat fold and parchment wrapping help the sandwich hold together, but they do not control temperature. Pack with suitable cooling and assemble the final wet components when you are ready to sit down.