Do I have to train the new owner even if I'm worried about my own job?

In most cases, yes — training or knowledge transfer during a transition is a reasonable part of your job, and refusing outright can create real risk for you even if your worry about your own job security is completely understandable. As long as you're still employed and being asked to do something reasonably connected to your role — showing someone a process, handing off client relationships, documenting procedures — that generally falls within an employer's ordinary right to direct your work, whether the person asking is your existing manager or a new owner's representative.

Refusing a reasonable work direction can itself expose you to consequences, separate from whatever happens with your job security more broadly. Your anxiety about the situation is better addressed directly than through passive resistance: it's entirely fair to ask the new owner or your existing manager, plainly, what the plan is for your role once the transition is complete, rather than guessing based on what you're being asked to do day to day.

If you do end up losing your job after training your replacement, that doesn't reduce your entitlements — you're still owed everything you'd otherwise be owed on termination, and helping through a transition doesn't waive anything.