Yes, you're right
To be more precise, 찜질방 has 사우나 inside it.
찜질방 has 사우나, 목욕탕(public bath house), PC방(internet center), 매점(snack shop), and all sorts of facilities.
Cheap ones cost around 5,000 for as long as you wish to stay, but some decent ones cost around 8,000 ~ 10,000 and you can actually sleep over in 찜질방 if you don't have a place to stay over the night.
I've been to 찜질방 twice - once with my family and once with my friends, and when I went with my family I stayed there for about 4 hours, and it was very boring to me. But all the other people seemed to be enjoying watching TV there