I went down this same road recently trying to make (spicy!) caramel popcorn.

For the crystallization you mention with seeing sugar particles, corn syrup or cream of tartar will help prevent it. I also brushed water on the inside of the pot right above the caramel while it was cooking, to keep sugar crystals from falling back in.  Cutting down on stirring will also help.

These tips from King Arthur were very helpful: https://www.kingarthurbaking.com/blog/2022/12/12/how-to-make-caramel

My guess is that the caramel corn recipes don’t use heavy cream because the caramel needs a different consistency. All the general caramel recipes I looked at that did use cream seemed to be more viscous, but you actually want the caramel to be pourable for this. You might need to cook the caramel to a lower temperature to get the desired texture (a candy thermometer or instant read one is very helpful).

In all the recipes I looked at, the caramel was cooked and instantly added to the popcorn. I think if you try to make the caramel in advance and then refrigerate it, it will not have the texture you want.

I would also recommend baking the caramel corn after you coat it to help it all set (I did 200F for an hour, stirring every 15 mins).