Other strategies that could come in handy before completely shutting down the API:

- Rate-limit the API, with increasing aggressiveness until you're down to 0 requests per unit of time;

- Introduce latency in serving the requests (assuming your edge can handle the increased volume of open connections).

Both of these introduce gradual degradation of the old API, without outright killing the business functionality that recalcitrant customers are nevertheless reliant upon. It helps spur a bit of urgency to switch to the new API, while remaining nice. Many (enterprise) customers will wait until the last minute to switch: essentially they are having to put in work without any tangible feature gain—at least from their perspective.

Regardless of the strategy, one other point I'd add is to monitor actual use of the API; if important customers are still actively using the old API, it would be unwise to shut it off.