Here’s my $0.02 worth.
Ball up the chicken breasts. Take some cotton string and wrap it around the chicken breasts to tie 'em up so they’re not a thin, flat piece of meat. Or overlap them on a tray. What you are trying to accomplish is reducing the surface area for the moisture to evaporate from. Ever notice that sometimes the thin end of the meat is dry while the thicker end is moist?
Cook with lower heat for a longer period of time. I cook chicken breasts in my smoker quite often. They are almost always very moist (unless I totally forget about 'em). The smoker cooks with very low heat ~ 200 degrees. Water boils at 212 deg. The juice in the chicken is water. Boiling water is just water vaporizing very quickly (that’s the bubbles). Thus the lower heat means the juice won’t boil out of the meat as fast.
Bake them with a pan of water in the oven. The other thing I do when I smoke meat is I put a pan of water in the smoker. (almost all smokers come with a pan intended for this). That water gets hot and bathes the meat in steam as well as smoke. If the environment in the oven is humid, less water will evaporate out of the meat.
Maybe coat them in oil? This may defeat the purpose of a skinless chicken breast if you’re seeking a low-fat meat. But since oil and water won’t mix, the oil can form a barrier holding the water in.
Most of all - don’t overcook. It takes heat and time to dry out meat. The more heat, the less time. The more time the less heat. Lean meat is very sensitive to overcooking. The meat will go from well-done to overdone real quickly. Because higher temps cause water to boil off faster, when you cook at high temp the window of time that meat goes from well-done to dry is very short.
If you’re like my wife, you probably just need to get it through your head that raw meat isn’t inherently poisonous. You are probably exposing your family to greater risk by cooking healthy foods to a frazzle than by accepting the risk that they might occasionally bite into a less-than-done piece of meat. The reason that ruining healthy foods is indeed a health risk is because your family will tend to skimp on that, then 15 minutes after dinner they’re digging in a bag of chips.