From your description it does sound like the dreaded "tank slapper" - the "wiggling" bit in particular. Caused by out of true and/or balance wheels I've not yet had a problem with it on my bicycles, but remember it ...fondly... from my motorcycles. With the limited speed/weight of an motorized bicycle I dunno if it'd be the same - but on a motorcycle the symptoms are pretty constant, if the wobbling starts at say 80mph, it'll always
start at 80 - it'll get worse if you go faster - but won't stop till you drop down to 60mph or so, depends on variables - but it's always a lot less than the starting point.
The only other thing I can think of is tracking, your front and rear wheel's alignment with each other and the bike...