you are on dodgy ground there mate, lived in Dubai for 14 years, you do not need a visa to enter UAE with a UK passport, but Irish passport holders do, so do USA citizens unless they have relaxed the rules now that tourism is growing.
They will not permit you to enter with Isreal stamped in your passport you will be out on the next "silverbird"[plane].
What we used to do was get the Israel entry stamp put on a seperate paper which you could remove on leaving Israel.
I saw a business man refused entry because the initials on his briefcase read J.E.W [ his name was Williams] but they werent having it.
Speak to your nearest US embassy asap or if you are going on business then ask the company you are visiting what they can do to help, every company employs a "Mr Fixit", some are more influential than others.
best of luck, but just dont go on the off chance.
Easiest thing to do is get a new passport pronto.