There's no easy way to answer this kind of question historically, but in essence: You would need to find a discrete way to turn yourself in to the FBI, who handles counterintelligence in the domestic United States. Whether they would grant you asylum or not would depend on a lot of factors, including whether they believed you and what you could do for them. They have you entirely over a barrel and they know it. They will not help you for your own sake; it's all about what they get out of it. And the chances of you getting a room with President Eisenhower are nonexistent — there is nothing you would say that would convince them that this was a good idea. (Anything you can say to the President, you can say to them, and they will pass it on.) Even if you had built up a long relationship of trust over decades, your chances of meeting the President are pretty much zero. Sorry.

You have to understand that the FBI is going to assume that anyone who was once a Soviet spy is probably still a Soviet spy to some degree (and that all defectors are possibly double-agents). They are going to be dubious about everything you pass them, because passing disinformation as intelligence information is the oldest trick in the book (even if it there is some verifiable truth in it, they need to be able to distinguish truth from lies, and that's hard to do with a single source).

As an aside, were you asking about a slightly later period (~1979-2001), turning yourself into the FBI like this might be signing your own death sentence with the Soviets. Why? Because the Soviets had a mole in the FBI's counterintelligence program (Robert Hanssen), and he would have told them what you were doing, and if you weren't in protective custody they'd probably have you recalled to Moscow and that would be that. Any sufficiently paranoid Soviet spy in the 1960s would worry about such a possibility as well. So this would be a very risky endeavor no matter what.