Please post a photo. You can clip off the damaged/dying pitchers at the leaf tip. A photo will help make a better diagnosis.

It's not good using any groundwater to water carnivorous plants. Your best bet would be to use distilled or rainwater. Groundwater, such as that from a natural spring, contains dissolved minerals within it.

As for the damaged pitchers, as long as they're not completely brown, then I would just leave them for the time being. They will still continue to photosynthesize (which is still a carnivorous plant's primary source of energy), and help it grow newer pitchers to replace them with. Just as you said, there is already a new pitcher growing. Which is always a good sign. Your plant will recover, just give it some time, and plenty of sunshine.

These need full sun and must always be kept damp. Should put it in a shallow tray and keep with water. Put it in 8 hours or so of full sun.

Update: after switching to bottled water and clipping off the decaying pitchers, my plant had a growth spurt. :-) Two more young pitchers have joined the first, with more on the way. Thank you for the advice! I'm very happy my plant isn't dead!!