I am just a first year phd student so as far as I know, digestion of DNA with a restriction enzyme will always produce a 5´ phosphate (despite blunt end or sticky end), AP removes the 5' phosphate group preventing vector self-ligation. You need to use AP if you are using only one RE in order to clone your insert and if your vector do not have blue-white screening option. Using two different RE sites producing non-compatible ends is ideal but when you are cloning products regardless of direction of cloning, using AP reduces the number of false positives. But it might also reduce the chances of your insert getting ligated. Using blue white screening vectors is more appropriate while doing tricky ligation where AP hinders the ligation process. I would suggest proceed without using AP and if that gives you only false positives then use AP.
What do you mean by band intensity? I did not understand you. If you are saying that you used AP in your RE digestion reaction mixture (as thermofisher fast RE and fast AP have uses same buffer) and run this on gel and getting low intensity that means either you used less amount of starting plasmid amount or your digestion is not sufficient. AP treatment do not affect the digestion as you should do it at the end of he digestion only for 10 min.