Paula, I'm not finding your picture, getting an error message.

If they are sprouting from the base, can I assume it's been a while since you moved them? I hate to tell you where to cut without seeing them, but if some time has passed, it's generally safe to prune anything appearing dead above new growth. You can take a fingernail, or carefully with a pruning blade/knife make a small scraping along the bark, check for any life in the cambium layer underneath to know how far down a stem is still alive too.

As for why they have been damaged (it doesn't sound like they are dying if they are producing new shoots), are you continuing to water and have you mulched to conserve moisture?

The link worked OK for me. Here is a clickable version:
http://share.shutterfly.com/action/welcome?sid=8AcNmzlw1Ytmab&notag=1

The "rhododendron bush" just looks like a bunch of dead branches covered with lichens except for these bunches of new growth starting in many places. The new growth looks healthy, so my recommendation is to do nothing except to start mulching. Rhododendrons have shallow roots, so they don't compete well with grass and other plants. They also don't like their roots disturbed too much like they would be if you were to pull the weeds and grass. Hence, the best solution is to use mulch to smother the weeds and grass. This also conserves moisture in the soil and protects the roots from temperature extremes in summer and winter. Good mulches include pine straw (needles), pine bark, oak leaves, etc.

The branch you label as dead probably is dead. I tend to wait 12 months to see what will sprout, but you don't want things to sprout too late in summer or it won't harden off for winter. Hence, I would say that old branches that look completely dead on August 1 can be pruned off.

I don't know where in Alaska you are located, but if you have a harsh winter, I would protect these new sprouts from extremes in winter sun and winter wind. A burlap wind break would work nicely. Leave the top open. If you get snow, that is good protection. You shouldn't need the burlap wind break then.

Here is a link that might be useful: How to grow rhododendrons and azaleas.