Now for a reality stop on this run-away train. Poor ol' Rich is just looking for a little advice, and he's getting hammered. I agree, he may have reached out to the wrong gaggle group if his primary concern is frugality. For that, he should have sought counsel at misc.consumers.frugal-living. Being an affectionado of both, I suspect Rich's ultimate "fix" here is a cut-and-paste job of several recommendations made, plus the piste-de-resistance in frugality: move existing downspout from current location to other side of gutter. Talk about frugal! And you don't have to get permission from the town/ city nanny-boobs on how to properly dispose of your old downspout, though I have a good idea how to do that ...
Regarding termites, they don't care if the wood is dry or wet, they just care if it's wood. Carpenter ants, however, are an altogether different insect - they love damp wood. So it is important to keep the water away from the wood. Regarding ice damns, all the one's I've seen and read about come from poor/no insulation. But rot, for sure this is going on with Rich's gutter assuming it's attached to wood. If so, then mold, moss, and funky fungi are sure to follow, if not already firmly encamped. He's probably also brewing some mean mosquito mix - maybe some 3E and West Nile viruses. Also, depending on your house particulars, you may want to pay attention to the downspout outflow so it doesn't become house inflow.
So Rich, I agree you should take advantage of your windfall sloped roof-line. But, to fix this right and frugal, its probably best you hire a handyman to perform what Shane wrote because is doesn't sound like you are too handy, otherwise you would have already figured this out and done it. So be it, Rich, there's no shame in that, and it keeps guys like me with food on the table. I'm sure you do plenty of valuable things I can't do, either.
- Bob Stanley, Handy Man