How can I add 3,000+ answer choices to a dropdown field?

There shouldn't be an issue adding 3k+ options in a single dropdown field. Although the speed of loading your entire form may vary depending on how long it is. Here are two example forms with 3500 options in it:

https://www.jotform.com/73041868243962

https://www.jotform.com/73041498343962

Just add them on the OPTIONS tab of the DROPDOWN FIELD properties.