I would imagine that the credit card coming will have their collectors try to collect the money from, then if they can not collect they will turn over the account or sell it to a third party who will try to collect the money. If the amounts are sufficiently large they will probably file suit against you. Not paying your credit cards without some kind of exit plan is not a good idea
Note that I am not licensed to practice in your state. This is not intended as legal advice. Please check with an attorney in your state.
When you stop paying on any debt, the debt holder (in your case, credit card companies) can sue you for the amount, send it off to collection agencies, and attach your bank accounts. Also, when you stop paying, you will see a decline in your credit score and the delinquency will show on your credit report. This means that interest rates on other loans will increase and you will find it harder to get favorable terms on future loans.
It is true that collectors will call and you the accounts are likely to be sold, and that's what you want to happen. It is also true that you will probably be sued and the amount of the debt is almost irrelevant. I have seen debt buyers file suit for as little as $200.00 in Texas.
Don't lose hope, by the time the suit comes you are likely to have strong defenses which can allow you to win. Even if you lose, in Texas nobody gets your house (except the IRS and your mortgage company), nobody gets your wages (except the IRS and child support), nobody gets your car (except the finance company), and nobody gets your pension or SS benefits. Most people don't have anything more than those items. So, even if there is a judgment against you, the immediate danger is not so great and a settlement can be worked out.
As for damage to your remaining credit card accounts, what difference does it make? If you plan to stop paying one, you might as well stop paying them all unless the balances are really small and you pay them off with the money you save from the other unpaid cards.