See, this is where things get a little tricky.
T-Mobile and AT&T run GSM Networks (with SIM Cards).
Verizon and (formerly Sprint) are based on CDMA networks. With CDMA Networks, even though you are using a SIM Card, the ESN and IMEI MUST be registered on the network for it to work. Now, taking a CDMA device to GSM is tricky. It MUST be SIM Unlocked and no amount of Firmware Flashing is going to get you around this fact. You have to have it carrier unlocked and even then, there is no guarantee it will work well on a GSM Network...
Now, beware, there is techno-babble incoming...
Most flagship phones now are what are considered "World Phones", meaning they support multiple bands. This saves the manufacturer a lot of money by being able to just make one phone and carrier lock the SIM and just flash Carrier Bloat onto the devices to "brand" them.
Verizon uses the following bands:
4G/LTE: 2, 4, 5, 13, 46, 48 and 66
5G: n40, n261, n260, n2, n5 and n66
4G/LTE: 2, 4, 5, 12, 66, 71
5G: n41, n260, n261, n71
The Fold 3 (regardless of carrier version or unlocked) can be used on:
4G/LTE: 1(2100), 2(1900), 3(1800), 4(AWS-1), 5(850), 7(2600), 8(900), 12(700 a), 13(700 c), 14(700 PS), 18(800 Lower), 19(800 Upper), 20(800 DD), 25(1900+), 26(850+), 28(700 APT), 30(2300 WCS), 66(AWS-3), 71(600)
5G: n2, n5, n25, n41, n66, n71, n260, SA, NSA, Sub-6, mmWave
So, in THEORY, SIM Unlock the phone (this is a MUST), pop a new SIM in there and it should work.
As for the Flashing Firmware, that is just to get the carrier specific ROM for your device, and from what I have read around here and on XDA, you can flash any firmware (unlocked or carrier version) on your device AS LONG AS THE DEVICE MODELS MATCH WITH THE ROM!!
Sorry, this may have been more technical that you asked for, but wanted to make sure you had all the info needed to make an informed decision.