Deep frying vs air frying temperature equivalence

It's impossible to give you a conversion. Or if not impossible, at least exceedingly impractical, both for the person deriving the formula and for you when you try to apply it in your case.

First, let's take the issue of underdoneness only. To make sure the center is done, you have to ensure sufficient heat transfer. In deep frying, most of the energy arrives at the food via conduction from the oil to the food, and is proportional to mass and the specific heat coefficient. Not only is the specific heat of oil almost twice as high as that of air, but also the mass of oil in a deep fryer is many times the mass of air in an air fryer. This means that in an air fryer, you get much more heat through radiation than through conduction. Since it is not the same process (such that you can plug in only the variables that differ and come up with some kind of simple ratio), it is completely impractical for a home cook to try to derive a conversion formula between the heat transfer in their deep fryer and their air fryer. Generalizing it and giving it to all home cooks would be even more difficult, and it would depend on the geometry of the fryers and on the properties of the food surface anyway, leaving too big a margin of error. Bottom line: forget the formulas, cook until done, whenever that happens for your particular combination of cooking device, temperature setting, and food.

Your second point was the surface. If you want the crispy golden surface of fried food, this happens through an interaction of heat, fat and usually also starch. The starch is usually already present in the food, either because you are frying starchy items like potatoes, or because you have breaded or battered your frying goods. The air fryer provides the heat. What you still need is the oil. You will have to toss your items with a sufficient layer of oil, generously brush it on. Then you will get a fried crust.

A third point is that you may be trying it with prefab foods (at least I have rarely seen people make their own cheese sticks). Prefab foods are precisely engineered to work with exactly the suggested method of preparation. Other methods, while edible, will likely produce a noticeable difference. You may not be able to get the same results with an air fryer.