Really, this fix shouldn't be that hard. Set your bike up on a trainer. Move your saddle back a few mm. Sit on it. See if you can support your weight if you take your hands off the bars. It shouldn't take a crazy amount of core strength, though, yeah, you won't be able to do a lot without sitting upright. Just test to see if that feeling of falling on your face goes away. If not, move the saddle back some more. Try again. Do it until you find that spot. Then put a plumb bob on a string and with your foot at 3 o'clock, drop the line from about the center of your kneecap. If you're significantly far back (several cm) from the pedal axle, then you might need more of a total fit overhaul. If you're close to the pedal axle, then this is probably a good position to try out on the road.

If you've moved your saddle back a lot, then you're going to want to spin for a bit to see how comfortable the reach to the pedals is behind the knee. You might want to lower the seat a little bit to compensate. (not the same amount as the aft movement--try a couple mm at a time).