Sounds like the problem is that you're falling forward on the bike, putting too much weight on your hands and trying to compensate by locking your elbows. If that's the case, move your seat back. It sounds counterintuitive, but moving the seat back takes the weight off your hands.

With the seat setback properly, you should be able to take your hands off the drops and be able to hold that position without falling forward. If you move the seat back, you'll also have to lower the seat slightly.