Good luck with your fitness goals, but from all I've heard, Crossfit is extreme and often dangerous approach to fitness.
I have seen plenty of healthy people that were in good shape go into that program, and find it was too extreme for their needs, or they got injured doing one of their extreme workouts and had to give up exercising for several weeks until they healed.
If you have to endure the extreme training of Crossfit to be able to compete in the Tough Mudder, you might be better off pulling out of the race. Even if you can't get a refund, you are better off taking care of your health by avoiding extreme training that could put you in the hospital. Besides, there are plenty of "normal" races you can compete in without having to go to such extremes.
This article from the NY Times has some really good background info on Crossfit. You might want to check it out before you get any more involved with that organization.
Getting Fit, Even If It Kills You
http://www.nytimes.com/2005/12/22/fashio n/thursdaystyles/22Fitness.html?pagewa nted=all 1624 days ago