Are you using energy saving techniques like:
arching your back to transfer weight to your feet when the overhang is pushing your body out from the rock and you only have feet below the overhang
using your pectoral muscles to reach across your body to grab a higher hold when you've lock off an arm rather than hanging from a bent arm
working your feet up high when you have a good under-cling available
searching for stemming opportunities to get weight off of your arms
getting a high foot and rocking your weight over top of it (most handy after surmounting the overhang)
Do you ever go out to just boulder? Whether indoors or outdoors you will find yourself frequently climbing overhanging rock. This will make you stronger and more confident (which will help prevent overgripping and panic) when you find overhangs on lead.