Questions about shifting from a new road cyclist
Sounds like an issue with bike-fit. Try the following:


While on the bike and stationary, lean on the wall with elbow (of one arm). The other arm/hand reaches the hood, your body should be as close as possible to the normal riding position (on hood).
two feet on the pedal
Use your back muscle and gradually reduce the load on the hood. Is this possible?


If the answer is no, which is what I suspect, try:


invert the stem, if you have threadless headset. Keyword: 'stem threadless adjustment' for further details on the internet.
increase the (quill) stem height, if you have threaded headset. You often seen this type on older bicycle (1980s and older), which is likely to be your case. Keyword: 'quill stem adjustment' for further details on the internet.
change the seat angle: change the angle so that the seat prevent you from sliding forward
change the seat position: slide the seat forward and have a more up right riding position. However, your centre of mass also shifts forward and your body is more likely to slide forward. Thus there should be a balance between seat angle and seat position (and stem length/angle).