My winter bike is scaring me

Ok. I know that I have a pretty low tolerance for strange sounds or movements associated with bike-riding. During my longer rides in the summer I would freak out about rubbing sounds and swishing sounds or regular clicking noises. During these times, I was calmed by a quick check of everything.

This Ross Compact that I bought as a winter bike is scaring me. I've ridden it about 16 miles so far. When I get up to anything over 10 mph on the greenway, it starts slightly twisting and pulsing. My seat moves slightly up and down. I feel like I'm on some kind of Dr. Seus-invented machine. I'm convinced that I will end up with a broken frame or something. My veteran cyclist co-worker suggested that it was the head bearing. The head bearing doesn't seem loose or anything. Isn't that a diagnostic detail? My dad wants to sell me his Da Hon Boardwalk folding bike. I am going to research this purchase as thoroughly as anything else. Apparently the frame hinge can get rusty really easily and sieze up. This was reported by a gentleman in England. I hope to use it as my foul weather bike because it has fenders.

In other bike news, my father has ordered the single passenger Rhoades Car a four-wheeled bike. He has been an avid cyclist for years and now he has Parkinson's so his neurologist has forbidden anything that is 2 or 3 wheeled. He is selling his car and will live car-free in a small town in North Dakota. When I was in high school I had a dream that my VW Rabbit had bicycle pedals and I was tooling around town on it.