HELP! Any suggestions? Any nearby city where there is nice lodging by the sea?
Your only other option would be to stay in Aqaba in Jordan. In fact, I wouldn't be surprised if it is a lot cheaper (I found Israel to be the most expensive country I have been to).
Heads up - make sure you read about any hotels on Trip Advisor before booking. A lot of the ones in Eilat (even the big name chain hotels) are quite bad from my research 2 years ago.