Contact company C and tell them that you have another offer pending (firm commitment from their side) and that you'd really like to accept their offer...but you need it in writing.
Once you have the offer from company C in writing (and not before) check it to make sure it meets the terms you thought you were agreeing to. If the agreement does outline what you thought you were agreeing to, and it still sounds like what you ultimately want, THEN it's time to decide what to do about company A.
How to deal with Company A is tricky. They've offered you a job, you've found other opportunities twice now, it's NOT a calm or easy situation. If company A offered you something different - more flexible schedule, slightly more money, a slight change in job duties - would that make company A's offer more attractive than Company C? If so, counterpropose those new terms to them and see if they're interested in keeping you on. Start negotiating, and have a definite idea of what you'll accept to stay with Company A.
If nothing that Company A says or does will entice you to stay with them...explain to them that you've been offered a fantastic opportunity, it really offers you better career options and lets you do what you love (or whatever your reasons for taking that role are,) and if possible - let them know of any people you know, who are both looking, and who might be a good fit.
(And, a slightly related question: which group of people did you get on with better? Company A, or Company C? Working with people whom you genuinely like personally and professionally, can make ALL the difference in the world. It can make a job with that's otherwise very good but NOT your ideal job description, into a much more positive experience. That might also factor into your considerations: do you really want to go with Company C over Company A? Why?)
Also: if you don't get an offer in writing from Company C...start the job at company A. I have had jobs offered verbally, with a start date and scope and everything...only to have them suddenly vaporize because of some unforseen internal factor. Do not count on Company C's offer as firm until it's in writing. Then look at all factors, and decide whether or not to take it.