Yep, looking forward to it. Tel Aviv is a proper Mediterranean city -and secular too, so there will be plenty to do on Shabbat (or even because it's Shabbat). Will look to stay in Jaffa if it isn't too pricey and getting around is convenient enough; otherwise stay near the beachfront (parts of Tel Aviv are pretty run down).
Beersheba looks really interesting. Literally travelling into the middle of the desert, an area that is recognised as a world heritage site and was a WWI battleground. As you say, it needs some research.