It sounds like you have "Aero" road brake levers which are designed to let you engage from above, "on the hoods" and from behind "in the drops". It takes a lot more force to be able to do this from above.

If you don't use the lower portion of your handlebars, you may consider changing to a different style of handlebar and switching to "flat" mountain bike levers.

If you have shifters built into the brake levers, I would agree that adding "interrupter" cyclocross style levers to the top of your handlebars is cheaper and easier than replacing your brake levers and shifters.

I just want to add that keeping your brakes adjusted correctly has a lot to do with this, also. Your brake pads should rest as close to the rims as possible, with a gap of a few mm, when the brake levers aren't engaged. Many of the repairs that I do to friend's brakes are because they simply are not adjusted correctly. Check this first, then go get those interrupter levers if it doesn't help. Those are fun.

The shims are good, but I also suggest that brakes are sometimes not as well adjusted/set up as they could be. Definitely a few things could help your problem including handlebar positioning. A lot of times bars are rotated up making the reach too long or putting the lever out of reach. A simple fit is probably what you need most!

I know Shimano road brake levers can be have their reach shortened by shims. The parts are cheap and just slip into place, but they might not be availabe for your year/make of levers. A trip to the bike shop is probably needed to make sure you get the right parts.