Please recommend a 100-300L or 100-400mm L lens

Sort of.. You want a lens with a larger aperture (or smaller f stop number.) The min. f/stop for the 2.8L zoom is 1:2.8 througouth the entire range of the lens, so the difference between the largest apertures through the total focal range is zero, which is not a very big range..

basically you want a lens that has the largest aperture for the focal length you want to commonly shoot at.

Lenses are commonly denoted by the smallest f stop at a given focal range (usually the smallest and largest focal lengths), since that's the hard part (and what makes them more expensive).. getting a larger fstop (smaller aperture) on any given lens is easy.

If you're having trouble visualizing the aperture process, here's a good animated gif of it.
http://www.dcviews.com​/tutors/tt51010.htm

"Because I'll be movnig around and can't lug a tripod around... have to take shots on the run."

IS is good at helping hand held shooting, but does nothing to stop the action at the subject end of the lens. So if you are taking freeze action shots, your shutter speed will need to be up for that and thus the IS is of limited value.

Subjects of slow action or still shots, give a great use for the IS and the slower shutter speeds. This also allows smaller aperatures for greater depth of field. An important consideration with telephoto shots.

I have 100-400L

I am new in photography (only 1 year taking pictures) and what I found out this last weekend, the 100-400L on low light (cloudy day for me on saturday) it is slow and high shutter speeds (to freeze action and have good clear shots) could not be reached unless I went up to 800 or 1600ISO (film speed). I decided not to increase the film speed to avoid noise, and I lost 80% of my shots. To fuzzy or not sharp enough. When the sun finally showed up, my pictures where sharp and clear. The ISO experience is something I learned that I will need to try in the future. I just ASSUMED that the quality of pictures would be that of my Digital Rebel, when I had it, and did not want the same experience. Of course, if you have money, there are programs that will remove the noise. Many people have attested to that and have informed me of increasing the ISO (to increase shutter speed) and use these programs. I will when I have money to get the programs. Now, I have the 50mm 1.8 and that worked out good when I took single shots of close ups worked nice. I shot in RAW and can manipulate the pictures for brightness and other parameters. Ok, now I will stop ramping about my experience. Get a 2.8 lens or lower. I had the Sigma 70-200 2.8 and liked it very much. It only had an issue when in Sports mode, would not find the subject therefore it would not focus correctly. It worked great for all other modes. When I went to exchange it for another, Uncle Sam was nice to me at that time and I got the 100-400 instead. Even though I had to purchase all my equipment again, I choose the 100-400 for its reach and beautiful pictures it produce in the daytime. I am now looking for a 2.8 or faster lens to go along with my 50mm. I will more than likely get the sigma 70-200 2.8 for close range or for the small soccer fields. I have heard the Canons 200 2.8 and the 135 2.0 lens is good to. this maybe the range you need for indoor martial arts. I hope this helps you and I did not go on and on and on. This is from my experience of each lens