Is there anything I can do-- other than keeping my head down and working really hard at independent training, letting my hard work speak for me-- to improve my chances of being assigned to a combat unit?
In the Guard you can, or will, enlist directly into a unit and will know this unit before you ship for Basic. You can, let's say a combat arms unit is 100 miles away, sign a waiver to make that drive for every drill, but that carries the additional time commitment of being there on time and no one cares about the distance because you signed a waiver. Something else to consider is a slot may or may not exist. You could have an infantry company 10 miles from your house (yay), the nearest AVAILABLE 68W slot in the infantry is 150 miles away (b00), but there's a 68W slot at a hospital unit 20 miles away. Decisions, decisions. The Air Guard and Reserves have great units that do aoermedical evacuation: basically transporting patients from large forward deployed hospitals back to places like Germany or the States.
At the risk of patronizing, I would encourage the latter option if available to you. The life won't be as harsh but you'll pick up some great knowledge and have a MUCH, MUCH better quality of life. Maybe you're thinking "screw that" and I respect the view, but understand there are other options out there. You could even find a Navy Reserve unit with Corpsman slots...you need to do this on your own. Hit up Google and search for army national guard units (your state goes here) or aeromedical evacuation units (your state goes here). Actually, I'll give you a free one, found using the methods above:
Aeromedical evacuation - Wikipedia, the free encyclopedia
Now start stringing together those key words and build a list of units, jobs, and locations. Narrow your focus and pull the trigger.