A Tapsigner is most frequently used with a software wallet like Nunchuck to create collaborative multisig wallets without having keys on the device, but if you have an nfc reader connected to your desktop then you can use it with your computer too. For cold storage of your main stack I’d say Coldcard is the way to go between the two.
You don’t need a Tapsigner to keep a Coldcard air gapped, you just need a microSD card, a microSD reader in your computer, and some way of putting power to the USB-C that doesn’t involve a computer (either plug directly into the wall or get the 9 volt to USB power thing Coinkite sells).
You can possibly use a Coldcard and Tapsigner together as part of a multisig wallet created using the Nunchuck app, since both products support nfc and your phone has an nfc reader. I wouldn’t do this with your first Coldcard though.
Coinkite has good guides on doing things air gapped, I’d recommend looking those up. I’ve found the best setup is to use an air gapped Coldcard in conjunction with Sparrow wallet on your computer, with the microSD card being used to pass the psbt and signed transaction back and forth. I was able to set that up without any issues by following their guide.
If all else fails ask nostr:npub1az9xj85cmxv8e9j9y80lvqp97crsqdu2fpu3srwthd99qfu9qsgstam8y8 directly.