How can I ask for these paychecks in a polite and professional manner?
I presume that you have always asked for anything work or business related in a polite and professional manner, so you already know how to do that. The issue in this case is rather what if you ask and they don't give you what they owe you? Which in this case is more likely than not. Don't waste any time and brain power torturing yourself (and getting us involved) looking for innovative and original ways of asking for what's owed to you in a polite and professional manner when the issue is most likely that they don't have the money to pay you.
What if they can't pay me all the paychecks upfront?
If in the US, contact your state's Department of Labor and review your options with them. Otherwise, check with whatever government agency is charged with protecting your rights as a worker in your country.
In fact, consult your state's Department of Labor right now and find out from them what evidence you should produce to show that you actually worked for your employer. I would not like to see your employer denying that you worked for them for the last six paycheck periods - and get away with the denial. Make sure that you assemble your evidence before you hand in any resignation. Also ask your state's Department of Labor if you are required in any way to give two weeks' notice in view of the fact that you haven't been getting paid.
When resigning, make sure that your resignation is in writing - this means, signed, date stamped by a notary public and in a PDF or on paper. And includes your reason for resigning i.e. that have worked since whatever date without getting paid and that you are owed six paychecks as of whatever your date of resignation is. Again, check with your state's Department of Labor if the PDF format is acceptable in a court of law. Oh, and demand in your letter of resignation that they reply with an acknowledgement that you have given your resignation. Religiously hang on to their written reply.
Can they still pay me after I have left?
You performed the work, you are owed the money. It's that simple. The fact that they owe you the money is absolute and not contingent on anything except you owing them money. But even then, they need to pay you first as taxes are collected on your wages. Irrespective of any money you might owe them.