Hi
No - Apple Watch Series 3 (GPS + Cellular) is not locked to any specific carrier and does not need to be bought from any specific carrier in order to be used with its service plans.
It has a built-in eSIM that copies information from the iPhone's SIM card.
The built-in cellular feature requires a carrier plan for your iPhone that specifically includes support for Apple Watch Series 3 (GPS + Cellular), including the carrier activating the device on the plan.
Follow the instructions here for setting up the cellular feature on your Apple Watch:
- Set up and use cellular on Apple Watch Series 3 (GPS + Cellular) - Apple Support
- Use Apple Watch with a cellular network—Apple Watch Series 3 (GPS + Cellular) only
If you still need help, then I suggest that, in the first instance, you contact your carrier again for assistance with setting up and using the built-in cellular feature.
Depending on your carrier's advice, I suggest that you next contact Apple Support (remote diagnostics and/or mail-in service may be available, if required), make a Genius Bar reservation or visit an Apple Authorised Service Provider for assistance: