I strongly suggest you post your CV on http://www.bayt.com/en/uae/
, The Middle EastÃ¢â‚¬â„¢s #1 Job Site, you can find a lot of teaching positions in the UAE. Once you register, make sure you have uploaded your CV online and apply actively to jobs that may be of interest to you. Also employers looking for qualified doctorate professors would be able to find you from the database if your CV is complete and active on Bayt.
I personally found Bayt.com to be very effective, not only did I get a lot of job offers through the site, but I also ended up working for Bayt itself!