How do I make a chicken cheesesteak that isn't dry?

Your chicken is dry because you over-cook it. 195°F is far too high for chicken, I cook chicken breasts until they are 165°F and thighs to 175°F, but with thin slices it's very hard to get an accurate temperature with a probe so I'd check for doneness by cutting through it.

Thin slices of chicken won't take more than a couple of minutes to cook thoroughly, if you put the slices in first by the time your peppers and onions are done the chicken's had all the goodness cooked out of it. Try cooking the peppers and onions first, then add the chicken when they are just getting done. Cook until they are just about done, then melt your cheese on.

To try get quick-cook chicken, without water leaking out… and not done like pencil erasers, you have to juggle, & juggle fast. Chicken cooks last.

The issue is that supermarket meat contains added water [legally]. As soon as the muscle fibres contract through heating, then the water squeezes out & you end up boiling it rather than frying it.
This can only be 'fixed' by not buying meat from supermarkets - but finding a good local butcher with 'known-abatoir-chain-of-custody' in an urban environment is not easy. If you do find one in a big city, they will have a constant stream of rich folks to their door. They will be expensive.
If you live out in the middle of nowhere, you may never even discover this problem exists.

So, assuming you're stuck with supermarket meat like the rest of us.

Get the peppers & onions going first.
When almost done, throw in the seasonings & cheese, preferably grated or pre-made sauce, stir briefly until just about heated/melted then turn out into a bowl. If you wait around for big slices to melt, you'll over-do your veg.
Get the pan as hot as you can & fry the chicken, fast & constantly moving - stir-fry style. You might need practise to gauge this accurately, but you can cook thinly sliced chicken in a minute [ask a Chinese wok chef].
When it's very nearly almost done, throw the veg/cheese mix back on top - quick stir & serve.
The chicken will continue to cook for the few seconds you're mixing & serving, but it won't get chance to start to 'leak'. Once it does, it's already over-cooked.

People are always so worried that chicken is going to kill them that they normally pre-emptively murder it before they serve it. It's no good having skinny quick-fry chicken slices reaching 165°F/75°C whilst it's still being blasted in the pan. It will be completely over-cooked by the time it reaches the table.

A dry chicken breast is overcooked. Some people actually like them that way which I personally don't understand but to each his own.

To avoid overcooking you cook it less - either by using less heat or less time.

Frying chicken breast (or indeed any piece of meat) is a matter of timing. You need practice to get it right. I personally don't have that timing instinct so I go by my (analog) wall clock. I watch the seconds hand go round the clock. To get a sense of timing do a test cook and cut the meat every minute or so to see if the insides have turned white. Once you know the timing cook the rest to roughly the same time, maybe add an extra 30 seconds or 1 minute because uncut meat take a bit longer to cook. But remember that the meat continues cooking while it's resting so sometimes all you need to do is let it cool down.

Experiment with your technique to get it right. Especially for cheap cuts of meat like chicken breast.

One other way of making this dish come together with more control is to fillet or pound flat the chicken and cook it flat, then after letting it cool to room temp, slice it up to add to your cooked veg in the pan for a quick toss before finishing with cheese under a lid without flame.

With practice on your stove this will give consistently great results.