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Sounds like wheel flop
https://en.wikipedia.org/wiki/Bicycle_and_motorcycle_geometry#Wheel_flop
doesn't look like you've got many options for fixing the geometry aside from a new fork...
Also the single tube heading to the front of the bike (and the dropped top tube, also) is going to make it really torsionally flexy which will definitely contribute to your steering problems. When you compare to something like this with the three tubes you can see how yours is going to have a way less rigid front end, which is going to feed back into the steering... maybe you could have someone add some tubes around the sides to give a more bucket-like shape at the front? And a really short stem and 800mm-wide bars might give you more leverage to control the front wheel also.
Some kinda update...
Tried yesterday to attach the shoe tree spring with hose clips, failed. Ordered a proper steerer dampener thing from scamazon. Should arrive tomorrow.
In the mean time today I attached a hollow aluminium tube to the front centre as a steer limiter, with the hose clips. It worked as expected as a proof of concept - all I had was the alum from a desk lamp, but for sure a rubber-coated steel would’ve worked best. I managed to get a half mile down the path to the shops with only one or two minor wobbles.
Then, going under the underpass the bike accelerated, the rod swung wildly speed wobble ensued and bam! I jumped off as it slammed to a halt.
The rod was jammed on the other side of the frame, the forks were 180’ the wrong way, and the alum tube was folded in half. Did not enjoy carrying the bike on my shoulder the whole way back - didn’t carry tools with me, stupidly, and the rod is impossible to pull back under the frame. Won’t make that mistake again.
Cycled the Giant Acid to run errands and thanks to the scrapyard I have a solid brass rod as a limiter. It will still bend if the force exerted on it is too great, but as a ‘guide’ it should limit accidental oversteer as proved by the sacrificial lamp.
Finally I removed the wooden bed and used some scrap elastic lace to act as bungee and hold the steering rod in place.
Of the three makeshift solutions I’m adopting, probably the elastic bungee is the key to a smooth ride.
Daren’t actually try it again until the dampener arrives tomorrow. I really hope it works.