Change Stroke Shape from Circle to Oval for Stroke Contrast

There's unfortunately no direct way to alter the shape of a stroke, beyond the end caps. 
There are a variety of techniques to create "brush strokes" of varying levels of realism, though most of these do get processor-intensive, as they're mostly based on either particle generators or heavy use of the Echo effect. 

You'll probably just have to import your Illustrator stroke, but depending on the complexity of your design, here's a trick that might work: 
If you make your stroke layer twice the width of your fill, apply the stroke, then non-uniformly scale it back to match, you'll be squeezing the stroke in the process. Here I've got a simple design made from shape layers, with a 600x600 circle fill, and 1200x600 elliptical stroke. 

If I then scale the layer to 50%, 100%, it'll smush the stroked copy back down to be the right visual size, but now everything on that layer is only half of it's correct width, giving me an elongated stroke. 

This is obviously a very simple example, but there are certainly ways you could apply this concept to more complex art, with the right approach. If you're working with vector art, taking advantage of shape layers' internal transform properties and structure could achieve this with more complex shapes. The Transform effect (likely using multiple instances of it on the same layer), should be able to get you a similar effect when you're not able to work with shape layers. 

You can take advantage of rotation or Skew properties (either within shape layers or the Transform effect) to angle the stroke as desired.