Hi Lisa,
I used to work at The Running Room, so here's my 2 cents. The Ghost 5 is a neutral shoe so you likely have arches that don't collapse inward, (which is a good thing in this case), so you could likely go with a cross trainer and be okay. However, for others who are reading this--one issue that you need to keep in mind is that if you do have arches that collapse or fall inward (overpronating) and don't wear shoes that offer the appropriate amount of arch support--you are taking the risk of hurting your feet/achilles/knees--and that is one reason why some people doing Insanity hurt themselves--and that is why many people who overpronate do in fact want to turn to a running shoe--b/c they are specifically designed to give you the appropriate amount of support to meet your specific needs.
As far as the 'side to side' motion--yeah, that's true, but regardless of what type of shoe you wear, you really need to be sure that you are working from the core at all times (core/hips/butt area) to prevent your knees and ankles taking the brunt of the impact. Not focusing on the core, and letting your ankles/knees take the brunt--will end up in injury--and shoes won't be the 100% solution to prevent that.
Also, remember, if you haven't worn the shoes outside, The Running Room offers a 30 day money back guarantee. (and sadly, good shoes are expensive.)
I actually wrote a post and created a video in my thread that addresses this issue, so drop on by to say hello and take a look. I've also included an article that I wrote for my coaching newsletter that goes into this stuff in more detail.
Click here to check it out.Best of luck and let us know what you go with!

Coach Zarno