Agreed. I'm primarily a mountain biker, but I'd still encourage you to first see if you can make drop bars work on that bike.

First thing I'd try is moving any stem spacers currently above the stem, to below the stem. If you're not familiar with loosening/tightening headsets, Google it. Very easy to screw up.

Consider rotating your handlebars a bit towards you so the hoods are a little higher (depending on where they are now). Find that spot where your hands naturally fall/relax.

If that doesn't fix things, I'd try something like a 110x25 Salsa stem to see if that gets you in a more comfy position. This might help: Stem Comparison Tool | yojimg.net

You can also hit up your bike shop for tips (they have the advantage of being able to see you on the bike). Stop by in the morning when they're slow.