I doubt that you will find any place on the beach but possibly a few on the cliffs who have a pool and security that want $1000 or less a month.
I usually just show up with the cash money and see what it will buy. May is a very dead month in Negril area and a lot of owners may be tempted by cold, hard cash. It always amazes me what people will agree to do when money flashes in front of their eyes.
Just my opinion.
Peace and Guidance