I'm late to this thread, but I think I have some insights. I think that if you "knew" how to no-hand, then you still do, and therefore the bike is preventing you due to some assymmetry that creates sideforces when none should be there.
I have three bikes, a Masi, a Mondonico, and a Trek (no, I do not have the chops to live up to their distinguished reputations...). All three will no-hand very well. The Masi has a short 95 cm wheelbase, about 58mm trail, and I had the frame aligned. The Mondonico (most twitchy) has a normal 99 cm wheelbase, about 52 mm trail, and has not been aligned, but I have paid strong attention to symmetrical wheel alignment and correct dish. The Trek has a longish 101 cm wheelbase, has been aligned, has a new headset, and has had the trail adjusted to about 57 mm. They are all from the early and mid '80s.
Before the Masi was aligned, it pulled while no-handed. After alignment, it no-handed without effort and seems to glide over bumps and edges. Same for the Mondonico after getting the wheels lined up right, but it's still a faster-responding machine. The Trek had a trashed headset, a bent fork, and a headtube-seattube twist, and it had only 45 mm trail. After alignment and fork adjustment I put in a new Chorus headset, and now it rides as if on rails, unless I want to turn. It is so easy to ride in traffic because it really is easy to go straight on it. I can even make cell phone calls on it (don't try this at home kids, only trained professionals were injured in this film ...).
So, I'd recommend first get your headset checked for free and smooth motion. Then get your wheels trued and dished properly and fit them in your frame as centrally as possible. If your handlebars are not centered in the stem, get them centered for balance. Make sure your handlebar stem is pointing forward. If all this does not help you find balance, perhaps the frame is in need of alignment. My order of operations is headset, wheels, balance, then frame. After the frame is fixed, the bike is about as good as a bike can be in terms of balance, so then it can only be you. But remember, you knew how to do this once, right?
Ken the 50+ hot dog.