Would the intensity of HIIT and endurance of cross-country cardio be useful for significantly increasing aerobic capacity to get to running a mile continuously?
Instead of grinding out a slow 11 minute mile try running at a 9 minute pace for a shorter distance. Then rest for a few minutes and repeat several times. Do that a few times per week and gradually increase the distance you run at the 9 minute per mile pace. Eventually you'll get to a full mile at your target pace.
Also if you're overweight you're going to struggle with running and be at higher risk of injury. If that's your situation then stick to lower impact sports until you drop the extra weight, then ramp up the running.
I'm in the process of losing 40 lbs (17%, cut) and then rebuilding muscle mass.