What power adapter do I need with my series 7 Apple Watch , since Apple only ships it with the cable

One option: Buy 20W USB-C Power Adapter - Apple

just bought Apple Watch series 7 SE. magnetic charging cable came w/it is USB C fast charger cable. Does not fit into power adapters I have using old style USB ends. Needs a smaller thinner end. Is the 20W USB-C power adapter by apple one that is recommended? Nothing confirms this anywhere on apple website nor is it on the apple store

You can use any USB-C port on a personal computer, or the charging brick from a recent iPhone, iPad, or MacBook Pro

You can use an older Apple Watch charging puck and cable.

You can purchase any vendor's USB-C charging brick from anywhere, Best Buy, Staples, Amazon, etc...

Yes -- the 20W USB-C power adapter or any other USB-C power source.