That's a neutral grip, if I had to guess without seeing it. You can try strengthening it, might have better results than closing your stance. It's all going to require practice and repetition, though.
If you're worried about losing a $5 ball for a particular shot, stop using $5 golf balls entirely. You'll know when you've moved into the type of game where the type of ball you play matters. I consider myself a fairly decent golfer and there's not a chance in hell I'm spending $60 a dozen. I do play the ProV1, but it's mostly because I find 'em all over the course that my work league plays at
If I'm buying, I like the Nike Mojo ($20 for 2 dozen) a lot. The noodle is also a good ball, despite its ridiculous name. Pinnacle Gold are just fine too. I've never felt I was using the wrong ball for a particular type of shot. Until you're really good, your scoring comes from play around the green and putting, and no ball fixes a bad short game.
Just stick to a lower cost ball until you're there. You need confidence in every shot, and worrying about losing your ball shouldn't even be on your mind at approach. If it is, play with disposables and remove one more mental piece. Seriously, this could be one instance where playing with the lost bin in my garage would lower your score.