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Sure they do, TheDonF. ESPN shows the national one every year, and there are quite a few local ones, semifinals, etc., that get shown on community-access channels and whatnot.
Shameless self-comment: I was the runner-up in the Maine state spelling bee in 1981. I won an AM/FM stereo with tape deck and turntable! All of my friends agreed that I had done much better than the winner, who won a trip to Washington to compete in the nationals and was summarily bounced out in the early rounds.
I finished 4th in the Texas state spelling bee in, oh, maybe 1988? I got an unabridged dictionary, which to this day is still the only dictionary I've ever owned. (The 3rd place kid got a TV. Yes, I was envious.)
Not only does ESPN cover most of the national spelling bee, one of the major networks (ABC) has aired the finals live in prime time two years in a row. I didn't watch live, but enjoyed watching on TiVo later (although it ran over and we didn't get to see the end, ahh! stupid TiVo).