I do a lot of stumpshooting, have for years; my favorite form of practice. It's so simple... but there's little things you learn around the way.
First off, you want to use the right points, and that's either judo's (by far my favorite) or blunts (I like the Ace hex blunts, but still use judo's). Field points would be a royal pain, as you're finding out. Even with judo's, they still bury in a stump on occasion; a sturdy fixed blade knife is my tool of choice to dig 'em out. I consider the knife a necessity, along with the bow, arrows, and glove. Blunts can sometimes stick, but not often. I don't use blunts much because I break more arrows with 'em, when hitting harder stumps or whatever. Judo's absorb some impact by way of the narrower point penetrating, and the spring arms help with this as well. With the blunts, pretty much the full impact is transmitted to the arrow.
But the blunts are a whole lot cheaper, and some arrows are gonna get lost - and sometimes just the tip and point break off and disappear.
As far as choosing your stumps (and there are other targets out there - leaves, pine cones, stray beer cans, flowering weeds, etc etc), well that's an ongoing exercise, and part of the fun. Basically, I'll just say I try to judge 'em so as to avoid the hard ones (arrow busters), trying for the ones that look older and punkier.
One other key thing is to consider where the arrow will go if you miss - ricocheting thru trees or limbs is another easy way to bust or lose 'em; and definitely try to stay clear of rocks!
Lots more tips, but I'll let others join in on the fun.