Take this very seriously. Sorry if this seems long winded, but the story is necessary to make my point. I feel stupid about the mistake I made but I also realise I am less stupid now than I was before it happened.
After I first rebuilt my Cobra, including installation of a brand new front suspension and rack, there was a very small bit of play in the steering. Apparently the threads on the steering shaft at the steering wheel end were damaged and didn't allow the nut to tighten all the way down on the splined adapter. I always meant to address this but I saw it as more of an annoyance, and fixing it it was a little way down on the priority list. I used to tighten it up every once in a while and at times it seemed like it may have been getting a bit worse but complacency set in and I began to get accustomed to it.
Briefly, one day, I noticed a "squirrely" sensation very similar to what you describe. Afterwards, I checked all the fasteners in the front and rear suspension plus the shock mounts, control arms, steering shaft u-joints, wheel nuts and tire pressures. Nothing was out of spec so, against my better judgement, I concluded that it might have been either my imagination or something to do with the road conditions at the time.
Next day, heading out to dinner with my wife, some friends of my lil' brother pulled up behind us in a gussied-up Supra that they had been trash talking about. We were on a good, quiet, open stretch of road, so I decided to teach them some discipline
. The Cobra bellowed and smoked the tires as I pulled away hard and held a good lead before it was time for both of us to start slowing down as we approached an intersection marking entry into a busier traffic area. They loved the rare display of the Cobra's fury and honked, hooted and waved as they exited and carried on.
Immediately after we turned left and accelerated slightly, the steering felt really strange, but went back to normal as we picked up a bit of speed. I slowed again and this time the steering was gone completely at idle speed -
TURNING THE WHEEL HAD ALMOST NO EFFECT ON THE CAR'S DIRECTION!! I was barely able to guide the car to the shoulder. At a dead stop, the steering wheel turned freely on the shaft 90 degrees in either direction. The splines in the aluminum steering wheel adapter were almost completely worn flat from the constant vibration and back and forth movement allowed by the loose fit.
Luckily, I had a pair of visegrips in the toolkit that I clamped on to the shaft after removing the steering wheel and I was able to limp us home using them to steer. Even more luckily, the steering hadn't chosen to let go at a time and place and speed that would have had some serious consequences.
The moral of this long diatribe: Do not take ANY loose or vague feeling in in your steering lightly. Being unable to control the direction of my car while it is moving is something I never, ever want to experience again!