Several things can cause this and some have already been mentioned. This bike has a different seating position than the EX-250. The 250 is more like a standard with upright seating position and the 600 has a more aggressive position. Because of this, you are likely gripping the bars way too tight, wrists folded back, arms straight or even locked at the elbow, back arched inverted (butt out), with your upper body weight being supported by your arms. Reverse every one of those things and it will stop.
Really, that's what you need to do. You need to get your weight off the bars (your arms) and get it back to the seat and legs. Bend your elbows slightly and loosen your grip on the bars. A good check to see if you're too tight will be the ability to flap your arms at the elbows. This will force the rest of the body positioning to become more correct and is critical to getting the bike to handle. Aside from the numbing of the hands, the original seating position described will prevent the bike from ever handling like it should. It will always feel tight in the bends and twitchy from unintentional steering inputs caused by bumps being transmitted to the bars from the stiff posture.
I assume you are wearing appropriate gear for that time that will eventually come, when you go down. With that covered, get the two Keith Code books, Twist of the Wrist I and II, and study them and practice what's taught. Track school is the best place to get the training and safe practice area but without that opportunity, patience and caution on the street is the next best order of the day. With a lot of studying and seat time, it will come together. Practice proper braking technique, too. In the meanwhile, don't get caught up in squidly speed sessions with the local gang who is likely riding with the same posture and all of the same ill handling problems that dictates. You're an accident looking for a place to happen in that mode. Safety first, then work on technique which will convert to speed. Never ride at a level that is scaring you or you WILL crash. The story will be completely familiar... in too hot, hit the brakes, went off, etc. etc... which is 99% of the time B.S. Panicking and getting tight, or being tight in the first place, causes the discomfort you described but more importantly, prevents the bike from handling and performing as it can. Go slowly, study and work on technique, the speed will take care of itself. Hope that helps... hope it wasn't too preachy.