the mesh tape I feel would be better, but after your comment on walking up there and the ceiling flexing, nothing will hold up to that treatment.
Also, blown texture might have been directly blown over the first stage of taping and then the moisture of Popcorn Ceilings could have caused the tape to just release. Popcorn is a good cover up for all defects in the taping and beding process.
Adheasive for tape is the mud itself. No special glue or anything, just mud. The tape is embedded in the mud with the trowel floating over the mesh.
If you do this taping and bedding work, you cannot go up in the attic for a good week until this all cures out. Are you attic ceiling joists 2x4 or 2x6's? You might consider your attic weight situation on those joists. With the weight of the stuff, the weight of the decking, then your weight, you might have that ceiling pancake down some day. I am not kidding. I would begin shifting all that weight of stuff in the attic to areas that are above walls that are down stairs, this will support more weight and be much safer.