Shalom Lisa!
Hammat Gader is in the north and very commercialized - I peeked in last year - just big modern swimming pools. The Roman ruins are inaccessible . . . mostly I think because there's no money for repairs and maintenance. Most of the hot springs are around Tiberias. I don't know of any hot springs near Ein Gedi. There are some natural springs. Ein Feshkha comes to mind, though I haven't been there in decades. There are cold springs that you can dip in on the Ein Gedi hike. Don't miss the synagogue ruins nearby that are part of the park.
I can't think of anything else around the Dead Sea. I will share my secret place with you, now that we are chaverot. When you drive down to the Dead Sea from Jerusalem, after you turn south on Route 90 you will come to a check point. Just before you go past it, there's a small road to the right. If you take it, you'll go up some switchbacks and get to a helipad (no longer in use). Park there for the best views of the Dead Sea. It's about a 10 minute drive.
There's tons of stuff to see and do in Jerusalem. A few things that are new that I recommend -
1. The Kotel Tunnel Tour. I've done it twice and would do it again. Buy tickets online in advance, the English tours almost always sell out, often well in advance - not open on Shabbat or chag.
2. The Israel Museum was completed renovated several years ago and it's superb, with a very good audio guide. The Dead Sea Scrolls (under the white dome) is the same, but there is a good intro video. The Second Temple Model (that used to be at the Holy Land Hotel in Bayit veGan) is now at the museum - really good with the audio guide. Then there's the main building with three wings, the archaeology and Jewish wings are not to be missed. The third is the art wing - also good, of course. The museum is open from 10-5 on Shabbat and holidays.
3. Ben Yehuda St from Kikar Zion to King George is a pedestrian mall. It comes alive on Motzei Shabbat, good place for souvenirs (bargaining is recommended, though it doesn't always work), street entertainment, cafes and street food, and people-watching.
4. The Jewish Quarter of the Old City, especially the Cardo from Second Temple times.