I installed the cheap 1/4" "cardboard type paneling over sheetrock using paneling nails and dabs of construction adhesive. The paneling is bowed throughout the basement project. What did I do wrong and how do I fix it. Should I not have used glue? Should I remove it and start over? All the seams are OK but the middle of almost all the panels are wavy. Some are bowed at least 1/2" gap against the sheetrock.

What did you do wrong, you bought cheap panelling to start and it was not acclaimated to the moisture level of your space. It is not likely repairable. Buy new, better paneling and allow it to sit in the room for 3 days. Glue all studs full length and nail every 6" on each stud, not just the perimeter