The way to get weight off the handlebars is to get yourself in a more upright seating position. Many of the more aggressive geometry bikes, and that includes most bikes sold in the USA, have a relatively steep seat tube angle. That will push the rider forward. It does put the rider in a more aerodynamic position but there's a price to pay for that, and one of them is more weight placed on the hands.

Getting weight off the handlebars is a two prong approach. First is moving the seat farther back. You can also get seat steams that are more angled back if you can't get enough with the seat alone. This will effectively place you farther back from the crank so you won't be thrown into the handlebars (or at least as much). Next you have to be able to comfortable reach the handlebars without leaning to far forward. That could mean a new handlebar with more sweep back. There are lots of choices on the market and the switch is easy. Hopefully you won't have to lengthen the cables which can complicate things but it shouldn't be a deal breaker.