I would remove the installed cartridges one at a time and shake them gently before reinserting, and then try a nozzle check.
Or replace the installed cartridges with the unopened one, but do shake them before trying.
In my 3800 I have successfully used many carts that were past the "use by" date by a few years and never had a problem. Yours are older than any I have used, but I doubt if a quick trial print would kill the whole printer, unless the printer itself is already badly clogged.