How to fix the caramel sauce

Okay. Yes you can save your sauce if it's crystallized without seriously impacting flavor. You can always start over with sugar, so to speak. Unless you burn the whole dang pot, candy can always be saved.

First of all, take your caramel (assuming you've tasted it and it tastes okay) and put it back in a pot with a fair amount of water. How much isn't important really because you'll be boiling it all off. But you do need enough to dissolve everything, and get all those wannabe fudge crystals back in solution. It will take some time to boil off and because of the butter and milk in there you will want to stir occasionally to make sure it doesn't burn. Because burning is the only real way to ruin things.

Stirring can indeed cause crystallization, but its not instantaneous and frankly it's... Not even a problem? I made walnut caramels for Christmas and they all decided to crystallize and turn into fudge after a couple days. They're still delicious. So there is literally no reason to throw away a batch of candy that's crystallized. Even if you can't turn it into what you wanted, it's still good stuff that's nicer than what you'll get at a store. (I am hurt by how many stories I hear of "ruined" candy being thrown out.)

But how do you keep crystallization from happening here? There are several methods to help prevent crystallization. People attach supernatural importance to "not stirring" but... I stir my caramels a lot and don't get crystallization. Stirring won't help you as much as reducing the heat and rotating your pot to prevent hot burny spots, but if it's stir or burn, stir, friend. Stir with my blessing.

That said, the more reliable method to prevent crystallization (where I am) is to add some corn syrup. Literally just add a tablespoon or two of corn syrup to your new pot of dissolved caramel liquid, cook it up to the consistency you want, and you probably won't get crystals again.

You can use any inverted sugar syrup in place of corn syrup though. So glucose syrup and rice syrup and golden syrup and honey (and more) would all work to keep it from crystallizing, but they all have their own character both texturally and flavor wise, so you might want to play around. Honey might be the best texturally, but has a pronounced flavor. Corn syrup is second best texturally in my opinion, but has a very mild flavor. Rice syrup has a lovely maltiness but tends to be stiff and stretchy.... So, playing around is a good idea here.

You could also add a little acid to the mix, but I would say maybe a teaspoon of lemon or preferably a pinch of cream of tartar. You will taste the addition of acid, but in a small enough amount, it would add a tang reminiscent of dulce de leche. Just be careful since you won't be able to take the acid back out.

The other thing I would emphasize is not to heat too high. Use the pot with the thickest bottom you have to try and prevent hot spots, and be willing to use a lower heat and take more time. If you have a glass top stove, you may well have more trouble with crystallization because the glass top doesn't actually keep a good steady heat. A coil top stove or open flame are both better if they're an option, but you'll probably want to stay on low or medium low heat with them. If you use a lower temperature, you'll be able to avoid stirring as well, for any good that actually does.

In short: you can always fix candy. Adding some corn syrup helps prevent crystallization. Acid does too but affects the flavor more obviously. Use a lower heat to prevent hot spots and avoid stirring if you can. If it's stir or burn, just stir. Haters gonna hate. If it stubbornly insists on crystallizing, so what? Let it be fudgey. Dissolve it in coffee. It's still good.