First I'm going to echo those who advise not chasing alone. If you need help asking about when to chase, and think you can limit yourself to one state, clearly you need more overall knowledge in order to chase safely.
On to your question. I like the second half of May but I tend to think of the three week period of mid May through the first week of June as having pretty much equal opportunities for success. Usually my trips start and end on a weekend to maximize available time, so that dictates the exact dates. If I'm already home before June has even begun, I always feel like it's too early to call it a season. For me the ideal (assuming I only have two weeks) is a trip that starts around May 19 or 20 so that I can include the first few days of June, up until around the 4th or 5th. Be prepared for possible disappointment in any week, even if it's a statistical peak climatologically speaking. You're always going to miss stuff before and after, because the season is simply longer than most of us out-of-towners can take a chase vacation.
Todd raises great points about staying on paved roads. But there's always a calculus, I mean if the next paved road is 10 or 15 miles away, I'm using the dirt road. There's a certain feeling of adventure somehow too, it adds to the experience in my opinion and can be a quieter and more picturesque viewing location. But after ending a chase in a ditch one year, I will never do it if the road is wet or has a chance to become wet, and I wouldn't do it without an SUV.
You cannot limit your territory to one state. Why would you, if you're on a chase vacation? And even if you wanted to pick just one state, don't pick Oklahoma, there's no place worse for encountering a circus of chasers on the road.
I agree with the spirit of what Todd said, you're out there to chase, so do what you have to do and don't squander opportunities. But at least for me personally, there is still a calculus there, I always weigh the length of the drive against the probability of success (success meaning supercells, not necessarily tornados). I mean there's got to be some sort of reasonable cost benefit analysis, I'm not going to drive from Kansas to South Dakota for a marginal opportunity, especially if it's a one-and-done and I'm going to have to drive all the way back down to Kansas the next day.
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