I've also been told that they would verify it before, so that would be out.

The law also requires that a pay range be provided by the employer upon request. I'm completely in favor of this law. Those who are strong negotiators will still be able to negotiate how they wish, and those who are in less fortunate positions won't have their pasts come back to bite them.

Why should you have to do that? If they can't ask, you don't have to lie, and wouldn't that be a better world to live in?

Is there any reason that you can't continue to do that?

Because your current employer may report salary information to one of those data brokers like Equifax and your prospective employer requests that information for verification. Lying on a job application is a good way to get get denied the position.