"You see rats with wings, but I see the internet"
I am here to show you yet another reason why "What if the internet goes down" is such a ridiculous question to ask. The internet is a network of networks, and networks do not have to involve computers (that's merely the most effective way we've found to create networks so far). Yes, you could even do the internet protocol over US Mail "snail mail", but today, lets look at the "IP over Avian Carriers" protocol. https://en.wikipedia.org/wiki/IP_over_Avian_Carriers
Signing the Transaction
For this, you'll want to follow the Sparrow wallet guide for cold card:
https://sparrowwallet.com/docs/coldcard-wallet.html
You can generate the PBST file from Sparrow, import to cold card using the SD card and then to broadcast, you'll need to connect the SD card to a pigeon.
How Pigeon Carriers Find their Way
Whoever will be your transaction broadcaster, will need to keep the pigeon at their site from some time, feeding it and making it feel at home. When a pigeon is released, no matter where, they will return to this site where they know there is food.
Security?
A signed Bitcoin transaction uses asymmetrical encryption to sign. Once signed, a new transaction using your signature can not be forged, so the only thing you're trusting your broadcaster with, is to actually broadcast your transaction.
Conclusion
As you can see, the internet is a network of networks and these networks can be crafted in a variety of ways. While not the focus of this post, it is also possible to connect to peers using the Bitcoin core client using IP over Avian Carriers, to ensure you keep track of the longest chain and give you the power to check that your transaction was in fact broadcasted.