I rode cross country about seven years ago and being single I had to face most of those problems. For some bills like electric and telephone I found that you can pay in advance, so if you know your bill will run about $100 while you are gone then just prepay that bill by that amount.
I also only brought one credit card with me and that was one that was set up to automatically deduct full payment from a savings account I had, so I could use it as much as I needed and not worry about the bill not being paid. I also brought along a debit card for cash withdrawls.
If you have an on-line bill paying account set up that would also be very easy to do, you probably won't have too much trouble finding public computer terminals, especially in libraries.