The exact procedure would be a matter of state law.  But normally a nonprofit corporation will elect to dissolve, stop its ongoing activities, and spend some time winding up its operations.  Basically its only activity after shutting the doors will be the winding up work of collecting its assets, notifying creditors and paying liabilities, paying employees, cancelling licenses, filing various final tax returns and other reports, and closing all your bank accounts. 

If you elect to dissolve now, your board (or a small group thereof) could be designated as the group responsible for collecting the assets and paying the bills and filing final dissolution papers with the state (which in some states may be at the beginning of the process and in some states at the end of the process).  You can approve a plan of dissolution now that specifies the charity or charities you want to receive your net remaining assets.  As the random receipts arrive after you have paid your bills, you can transfer them to the proper ultimate beneficiaries.

When the receipts diminish and you get tired of waiting, you can try to assign your rights in the remaining payments to the beneficiary charities, especially if you know what is in the pipeline, and notify the anticipated payor(s) where to send the funds.  That may not work with all of the potential payments or may be ignored by some payors, but it would be a final act for your organization and your people could go on with their lives unencumbered by further corporate obligations.

The process is not a short one under any circumstances, but if you get help from a local lawyer or accountant familiar with the process you should be able to make it work for your situation.