Your brownies aren't more moist, but they are denser which makes them a bit gooey. A good gluten free flour these days is usually pretty comparable to plain all-purpose wheat flour when it comes to absorbing moisture, some GF flours absorb a bit more so if anything your brownies are more likely to be drier than when you use regular flour.
Gluten creates structure that traps gases that are created by leavening agents and the expansion of water vapor. Without that structure your brownies will rise but then fall back more after baking. This will make the result denser, the closer texture leading to that gooey mouth feel not everyone wants.
I personally like that texture, in fact after making my brownies GF for a co-worker I was told never to go back to regular flour with my brownies again. So it may be a hit and you don't need to do anything but sit back and bask in glory. But, if you do want a drier texture you could bake it a bit longer, add one less egg or a bit more flour. Xanthan gum is a popular gluten replacement, however I would not recommend it in your case as it will make the result more gooey which is opposite your intent. Your GF flour may have xanthan or other gums added which is fine, I just wouldn't add any more.