You wrote:
"But the idea of using cloth seems like it would give me that cold feeling against my skin"
Since I wear cloth diapers all the time at night, (Except when traveling...) I have a couple thoughts on this...
Regarding a cold wet feeling with cloth diapers - I guess it depends on how you sleep. I always have a blanket covering me, even in the summer. Wearing cloth diapers with properly fitting plastic pants, under a blanket, the diapers pretty much stay body temperature when wet. Most mornings I don't even know I've wet until I reach down and check my diaper under the waistband of the plastic pants. Now if you sleep with nothing covering you, then perhaps a cloth diaper might cool enough that you would notice it.
Also, plastic pants have a part in it too. If you splash water on your pants, part of the wetness you feel is the cooling of the water as it evaporates. The same way our bodies cool by sweat evaporating. Have you ever had a diaper leak while you were sitting down, and you didn't notice until you stood up? That's because the wet spot was kept at body temp, then you stood up, it started cooling, and you felt it. With plastic pants over the diaper, it stops the evaporation and stays warm longer. Note: I have used snap-on plastic pants a few times, and my diaper seemed to cool down faster, given the gaps around the snaps.