ModMic Uni 2+

My experience with headphones, headsets, etc is that you're best served by a) an external DAC and dedicated headphones and b) a separate, dedicated mic. There are a lot of super expensive lapel mics out there which might suit you but I used a modmic for years without issue. They have several options but the cheapest, analog option is probably fine. For sound quality/headphone choice you're going to have to choose between >open ear (probably not what you're looking for considering you and your wife are right next to each other) OR >closed ear headphones I've had good experience with Audiotechnica monitors but if you're looking for a more gamer-y EQ there are other decent options they offer. A decent setup that won't break the bank is a pair of closed audiotechnicas, analog modmic, and a schiit fulla external DAC. Should run you about 250 dollars-ish but I promise getting dedicated headphones instead of a headset will last you much longer and generally sound better. Also, any dedicated microphone software is likely going to be nearly useless. You're literally fighting the laws of physics trying to isolate your voice from your wife's. The only way I could feasibly see you doing this is with a microphone that is right next to your mouth. If you have Nvidia (I do not so I can't verify this) they have RTX voice which might offer some more robust cancellation/recognition. AMD has something similar I believe but I don't use it so YMMV. Most voice chat apps also have built in features to isolate noise. All you can really do is tune the sensitivity to be very, very low and hope for the best.

I run HD6XX's with an external mic (a modmic turns them into a great headset). I almost exclusively use them for gaming at this point, but they're phenomenal headphones as well. The HD58X's are excellent as well and easier to drive. Both headphones are around $200 on Drop, give or take sales they have going on. I'm sure they'll have something going on for black friday.

the 560s is good, just slap a antlion modmic on there and ur good to go

Started out with a $20 tonor mic but had connection issues with it, it would always drop out and I’ll beed plug/unplug several times. It also picked up my keystrokes. Went to Samson Q2U with a cheap arm, much better/clearer voice according to my friends. Also got a modmic which does the job but kinda of annoying to deal with the wire.

Dynamic cardiod mics are the way to go here.   They can be used with speakers and other people in the room, but you have to "eat the mic," or you'll be too quiet. The pickup range is short unless you crank the mic gain (volume). That's a feature, not a bug, and the reason they work on stages.  A pair of Behringer XM8500 (budget Shure SM58), and a pair of interfaces would be a viable path. There are USB to XLR cables with integrated interfaces, making them a super cheap way to go, albeit without a discrete gain control. The Samson Q2U is also viable for XLR and USB, but more expensive.  I don't recommend Antlion's ModMic for this setup. I tried it and it couldn't handle room rejection and still created an echo.