Just run more. More mileage, more often, with a little more intensity added in as you go.
You're currently running 10-15 miles per week, maybe hitting 20 once in a while - that's pretty easy and low mileage for someone who wants to be a competitive distance runner, and you don't need 3 or 4 days off per week to sustain that. Start logging just 2-3 miles on the days when you're currently not running at all, keeping one day per week reserved for total rest (zero miles) and your fitness will rapidly increase. Even if your mileage isn't increasing right away, just committing to a run every day (6 days per week, at least) will do you wonders.
Once you're at the point of running 5-6 days and 25-30 miles per week, start looking at adding 3-5 miles per week every single week. By the time you're in the high 30s or 40 miles per week, some magic will start to happen — your ability to handle harder training will increase, your speed will begin to increase by itself, your perceived effort level will drop across all distances and paces, etc. If you find yourself capable of logging 50mpw consistently in the weeks leading up to the cross country season, you'll be head and shoulders above most runners and probably be well into varsity territory.
Regarding that weekly interval session you mentioned - skip it for now. You really don't need speed workouts or track sessions in the summer before a cross country season. Just log the mileage, start adding a long run of 8-10 miles once per week (most people do theirs on weekends) and get to the point where you can run 12-15 miles at a stretch. Also start adding a couple quick "striders" of 50-100m to the end of your easiest runs, and work on getting to the point where you can finish an easy 5-mile run and add 6 to 10 striders in the final mile as you head toward home. That's all the speed work you need - just keep in touch with what it feels like to go fast once in a while, and let your coach dictate hard track workouts etc as the season proper gets underway, once you've put in the time and effort to build a strong aerobic base.