Eilat – Neptune, holiday inn (the one adjacent to the airport), i think also blue lagoon. They used to be mehadrin but i would suggest to verify before going. And i never stayed there for shabbos.
Dead Sea – doesnt have any mehadrin hotels but in some of them you can pre-order mehadrin meals and they’ll just warm it up for you double wrapped.
Most hotels have a full time mashgiach. You may want to ask him what you may eat. Often the lettuce is katif, and the vegetables he will maaser himself. The meat is often Beis Yosef shechita etc…