It sounds like you’re using a zoo med reptifogger. I had the same experience with the hose dripping frequently and the water collecting on the screen, but it can certainly be a good way to provide humidity. I also live in a very dry climate. To increase humidity, I use reptile prime substrate. It holds moisture well when misted. A humidity box, tupperware with a hole cut in the lid filled with moist sphangum moss, works really well for my snakes. It won’t increase the ambient humidity, but it gives them a nice humid spot to go soak some up.
It sounds as though your bulbs are currently making contact with the screen when placed in that dome? If this is the case, the dome is not deep enough for the lights you are using. I would recommend looking for a lamp with deeper dome. Check the wattage of the lamp and make sure you get the correct bulbs. I would agree with the suggestion above about switching to a ceramic heat emitter from personal experience, but make sure that you get a thermostat to regulate it if you do.
Then, you could get a reptile lamp stand so that your lights are hanging and are no longer making contact with the screen at all. This would prevent the same issue from happening again.