Wobbly/Unsteady Car

Hi,
I'v been driving my current car for a couple of months now and only
recently has it been feeling very unsteady while driving. When I'm driving it
feels as if one of the tyres is going to come loose. It's worse at higher
speeds as I take the motorway to work but even when reversing out of a
park I could feel it. It's not a constant vibration but more of a 'dipping' of the
car or shimmy. Last week I put air in the tyres and soon after is when the
problem occurred. At the time I had put it a higher PSI (36) so I thought it
could be that so I let the air out to the recommended 32 PSI but still no
change. Also was due for a check up at the tyre shop where they balanced
the wheels. They couldn't find anything else wrong and even after that
appointment the car was still unsteady. The tyres are very new as they
were recently purchased at said tyre shop back in November. The problem
must be quite obvious as well seeing as when I'm driving now people are
giving me a wide berth or are leaving a large gap which they didn't do
before. The car also seems harder to accelerate than usual and seems to
me to be using more fuel. Could this still be tyre related or is there an issue
with another part of my car? Any help is appreciated!