Should be fine

Agreed, no need to panic. =)

1. Run a ventilation fan in the area to help dry the pots out faster.
2. If you're very concerned and/or they are very water sensitive species, you can put the pots on some folded or layered paper towels, which will wick out some of the moisture.
3. You shouldn't have to remove your cacti, that will just damage the roots further.
4. If they were being kept in a very cool area, keep them in a more comfortable temperature area (does not have to be hot).

After repotting, I usually don't water for a week. Even then, I start watering very lightly and increase from there as the weeks progress further.

Not so much for cacti, but other succulents I pay close attention to the plant response when starting to water lightly. If the plant does not respond or begin to show growth, watering stays at that minimal level (sometimes it could just be a few mL of water).

This is important when potting up bare-root plants or those with extremely small root systems. Once the plant is growing, I tend to water quite freely without any worries.

Thanks for all the great and detailed information ewwmayo, will definitely follow your recommendations. :) They're in terracota pots so at least that will help with evaporation, I hope. I was so excited to finally get them out of that concrete peat and I don't want to ruin them by over watering.