First: Scrubs are good as newb garb, but they don't hold up so well to physical activity (as you've learned). Better to make your own pants out of a good, sturdy material, like worsted wool (which basically refuses to rip).
Second: When you sew them, as Agneezy said, you want a diamond gusset in the crotch. I'll see if I can find you a pic or a tutorial without trying to take explanatory pictures of my own (black, of course *eye roll*) pants...
Ah! Here:
I use Figure 2's method or making two tubes with a square (diamond?) gusset for the crotch. It works awesomely. You can even taper the legs a bit towards the ankles for more historicity/easier fit under leg armor. I've never had any problems with blowouts. (Note: the dimensions on the picture are for "braies" which are basically medieval boxer shorts; they'll come out pretty short, but are AWESOME to make out of linen and wear under tunics (without other pants!) or under another kind of pants (especially hosen). You'll have to figure out how long and wide you need the legs when you pattern your new pants.)
Third: Be sure you sew the seams right. Good backstitch and Simple seam finish (zig-zag and press open) will serve you fine, but go over the corners of the gusset twice with each of those stitches. Otherwise, French seams might work for the gusset, but I ain't gonna try to figure out how to make that work (yuck!). Make sure you give yourself plenty of crotch and plenty of butt room.
With the drawstring at the top, these pants can easily be made with rectangles for the legs that are each 3/4 of your waist measurement wide (resulting in pants with an almost one-and-a-half-of-your-waist-measurement-wide drawstring top!) and still work fine, if you need. Pants that baggy will look better with winingas (strips of fulled wool ~4" wide wrapped around the shins) or tucked into low boots; skinnier pants are more appropriate for most places during most periods (in Europe, that is).
I recommend making your first pair out of bedsheets (Goodwill or yard sales, SUPER cheap) for the practice and to get the sizing right, then order some nice worsted wool or heavy linen canvas (from fabricmartfabrics.com or fabric-store.com are your best bets; wait for sales!) for your real pair of fighting pants. It would benefit you greatly to make two pair of good pants, so when one starts stinking (less of a problem with wool than linen, and less with linen than cotton) at an event, you can switch out lol.
Good luck!