The short answer is that your intuition is correct, but spacesuits actually do have exhaust. Spacesuits include a layer that works like liquid cooling inside high end computers: water in tubes makes direct contact with skin, and circulates to a heat exchanger which vents ultimately the heat out of the system by evaporating water (or I think sublimating) into space (or else carries it back to the ship via a cord, and the ship deals with the heat as part of its own ventilation system).
If you were floating in space naked, you actually wouldn't freeze to death. The body radiates some heat but on earth, most of it leaves the body through sweat and convection — you heat up the air around you, which gets blown away so you heat up the new air around you which gets blown away, etc. Vacuum doesn't carry heat away, because there's no air to heat up. All kind of weird thermal shit would happen because of the zero pressure, but if you're in a suit then your body's moisture and gasses wouldn't be exposed to that zero pressure.
Just think about
thermoses. They keep coffee hot by surrounding it with vacuum. Likewise, a person in space is surrounded by vacuum. If a spacesuit was just a sealed tin can, you'd cook inside it eventually.
I don't think it'd get oven-hot. The equilibrium would happen when you die, which would probably happen long before oven temperatures.
