Dave, I feel your pain. I'd also second the advice for getting some sort of coring system. I personally have a McNaughton, which I have a love hate relationship with. I didn't love it as much when I got it as I do now. I've cored probably a couple hundred bowls on, and need to expand to the "jumbo" cutter or get a OneWay easy core for large blanks. I'm also a packrat for scraps. I cut all my own wood from arborists & friends blow-downs, etc, so I make my own blanks, then round them on a bandsaw. The corners are usually saved if there is any burl or interesting figure; otherwise, I have a chicken wire bin, 4x4x8' behind the shed which ends up being firewood during the winter....
There is a point though, where one realizes that sometimes its just not always worth it to finish turn the cores you remove. I don't know who said it, but there is an often repeated phrase: "life is too short to turn crappy wood", which is applicable here. Sometimes, I throw the whole core in the firewood bin if its going to be really shallow, or just not interesting. Then again, sometimes I get a wild hair and get a 4 or 5 bowl nested set, which is as many as I've removed from a blank before. Maybe I'll get into a 7 or 8 bowl set like the Mahoney BoxElder burls that he is so great at!!
Just know that any coring system has a learning curve. Don't put on any blanks that you're really fond of until you've practiced with several "junky" pieces first, or you'll make a really nice lampshade or funnel....