There's probably not many programs that can do this, otherwise it wouldn't have been such a pain at the time to solve the types of problems you're having.
You can't "probe" an IRQ to know what device is using it, you can only identify a device and then via some proprietary mechanism (maybe) figure out what settings it's using, or use PCI or ISA plug-n-play if the cards are new enough, which is just a standard way of doing the same thing.
Your best bet is to probably find some manufacturer configuration or setup program for both of your cards and run that to see if it will tell you the current configuration.
Otherwise you'll need a program that is aware of both those cards, so it knows how to query the resource info. Possibly the only one is an old version of Linux, if it includes drivers for both cards. If you can boot that and load the drivers, then you can query the kernel to see what hardware resources are in use. Note that 386 support was removed from Linux a while back, so it will have to be an older version.
I'm not aware of any DOS program that can do the same. There are a few that come close, like MSD (Microsoft Diagnostics) that came out with DOS 6, but like all programs it will only provide limited info, and possibly none at all about hardware it isn't specifically designed to detect.
I assume you've tried all the usual troubleshooting steps? Different slots, cards work fine one at a time, etc.