"then I doubt galaxies would have the shapes we know."

What would you expect they would look like, if they were?

More spherical?

Is there any reason to suspect that Sol's path and other stars are traveling in a nearly perfect circle?

In our solar system Mercury's Orbital Eccentricity is 0.205 and Pluto's is 0.244. Earth has Orbital Eccentricity of 0.017 and other than Venus and Neptune [0.007 and 0.011] it's path is one of those with a path most like a perfect circle.
http://nssdc.gsfc.nasa.gov/planetary/factsheet/

During Earth's yearly orbit it gets closer and further from the Sun by distances of millions of miles, but because of the large diameter of the orbit [slightly less than 300 million km] it would visually appear as nearly a perfect circle.

"The galactic disc, which bulges outward at the galactic center, has a diameter of between 70,000 and 100,000 light-years.[20] The distance from the Sun to the galactic center is now estimated at 26,000 ± 1400 light-years, while older estimates could put the Sun as far as 35,000 light-years from the central bulge."
http://en.wikipedia.org/wiki/Milky_Way

Therefore if the Sol system traveled in a nearly circle path as does Earth around the Sun, we would get further and nearer the galactic center by hundreds of lightyears.

But, how could we know Sol's orbit isn't like Mercury's and Pluto's. Or even a higher orbital eccentricity- such as similar to a comet's path?
And if Sol and most stars have a low eccentricity are there exceptions?

The things that make a galaxy not have stars mostly all [99%] travelling in low eccentritic orbits would be gravity interaction between stars, varying density of galactic space, supernova events, a history galaxic collisions, stars ejected from the galactic core, and probably other factors.