Squirrly is what I call this. I own a '97 E320 and it had exactly this
subtle steering behavior.

IMHO the tires on both our cars are too large and magnify such problems.
OK, here's why.

I found that the front wheels' toe in/out caused the squirrly steering.
As one drives on a local street, say at 25 mph, the car seems to favor
first one side then the other. IMHO this is caused by rises and dips in
the road acting on first one wheel then the other. If the front toe-in
or toe-out exceeds specification the car will self steer because the
wheels lack parallelism. Visualize too much toe-in - the leading edges
of the front wheels closer together than their rear edges (spec. is
about 1/8") - and a rise in the roadway on one side only, the wheel that
now must go up will steer the car in its direction vs. the direction the
other, now less loaded, wheel is pointed.

I check the front alignment thus: Exactly center the steering wheel and
drive into the garage or other flat surface stopping the car with the
parking brake. The steering wheel is dead center so each front wheel
also should be too. The body overhangs the wheels so one needs to use
spacers to compare the front to the rear wheel.  I hold a 2" wide bubble
level as a shim (not for its bubble but for its consistent width) on the
front alloy wheel and sight down to the rear wheel against which I
placed a Chevron Techron plastic bottle (not that that's special but it
gives a 2.5" space from the wheel). Using this simple method I've found
one wheel was toed out and caused the squirrly steering. One can
correctly calculate the exact width of the rear spacer using high school
geometry, adding the front spacer's width and adjusting for the car's
front vs. rear track widths. But for a quick check these figures will
suffice.

Hope this solves the problem.

Those tyres sound quite exotic. Is it possible that they are
unidirectional and are on the wrong way round?

I have same chassis as yours with same size tire... Avon Tech M500. I do not
have free play on steering wheel but I know my idler arm bushings are shot!

When I had stock Michelin 195-60R15 with stock wheel, talk about
squirrelling!!!! AUGH!!! My car goes everywhere it wants to go on highway...
constantly readjusting the steering.

I upgraded my wheels to the W211 style... OE Mercedes and new Avon 16"
tires.. What a difference! It tracked straight and sticks like a gum. Yet, I
still didn't change the idler arm bushing yet... I don't feel squirrelling
anymore. But before I got my car, the previous owner changed all the shocks
and also changed the front lower arms... so this might be your case...
bushings on lower control arm is bad. Other thing to check is your sway bar
linkage on all 4 corners.

usually this is a very slight alignment issue. The front wheels might
be ever so slightly towing out, so that the straight ahead position is
on the verge of turning.

I know you said that you have already had the car aligned and checked,  
but still...

These symptoms sounds EXACTLY like the problem that I had recently.

I checked everything on the suspension and steering and found the following:

Wheel alignment - OK
Control arms, ball joints, bushings, steering parts, etc. - OK.
A front coil spring was broken at its lower end.
Front sway bar (stabilizer bar) was worn (in the area under the bushings the
metal bar was thinner), and the rubber of the bushings was aged. This
resulted in too much free play of the sway bar.

First, the sway bar and the four rubber bushings were replaced by new parts.
And this actually solved completely the problem with the wheel wobbling.

The replacement of the both front coil springs after that was without
noticeable effect.

"" wrote:

Hi there

I experienced this with a 300CE. It was actually preceeded by some
creaking at the front end and which I took to be the front springs.
The creaking stopped but the centre line giddiness got worse and of
course I found out the problem and which is a common W124 weak point
when I drove over a sleeping policeman or ramp at low speed and the
lower ball joint pulled apart throwing the entire suspension strut
assy out to one side and grounding the car. These are a sealed for
life item and Merc do not give a fixed replacement interval rather a
service check interval. The problem though is that because of the
construction and the way the springe loads the whole assembly, any
play in the ball joint is not easile checked on a ramp with Pry bars
or levers. The only succesful way is the compress and unload the
spring with clamps. At this stage any noise or giddines from the front
end I replace them, they are only about $20.00 each.