Did you have a Home Inspection BEFORE you bought your dream home!? You say you FOUND your dream home, but does that mean you bought it anyway?! In the vast majority of states, termite inspections, alone, are required by the bank / mortgage-company.
Well, before you do anything, you need to educate yourself about how temites live and thrive. IE., become one. Do some reading. Prevention is a good start. Try Termites for starters. If you can eliminate or dramatically reduce the 'conditions' in which they live, you'll sleep better knowing that your home isn't being eaten, literally.
Once you've done this, THEN get the joists replaced / repaired / sistered. DON'T put the new joist right up against the rotted one. You run the risk of transfering the 'problem' from the old to the new. Have a pro do this work. Yes, you can really get rid of them. Just don't permit the conditions in which they thrive to exist in or around your home. Allow air flow around the home by removing / trimming back bushes and trees at least 3' from the house. Fix any gutter, downspout, and drainage problems. Simply put, let the sun and the air in!
If you need more, post up. My best to ya and hope this helps.