Adjunct to Full time position conflict
You can ask for an unsigned copy of the offer letter for full-time, and explain that you need to look it over so you can decide how to proceed with the adjunct work. They should understand and even appreciate that you don't want to leave anybody high and dry. Likewise, you might consider contacting the chairs of the departments where you are adjuncting and get a sense of how difficult it would be for them to find someone else to teach the course. It could go either way. They might not have a good back up plan, or they may have a reserve of graduate students who need to teach to improve their CVs.