I wouldn't be so fast to blame the threaded/quill system per se. Remove and inspect the stem. Then, make sure you do not have the quill any higher up than the min insertion/max height line and that it is really tight. Some quill/headset combos are looser or tighter than others. A new, better stem may be all you need.

If you go the threadless route, you'll need a new fork, headset and stem (and spacers maybe). It'd be ideal to match the rake of the new fork to the old. If you don't know the rake, you can probably eyeball them next to each other. I assume the old headset is the ISO standard, so any newer 1" threadless ISO headset should work.