My MAXX switched between 4G and 3G a lot as well, but it hasn't been as much of an issue with the Galaxy S3. With the Galaxy S3, it tends to stick with 3G in weak 4G areas (places where my MAXX, Galaxy Nexus, and Rezound would all bounce back and forth between 3G and 4G). This is done on purpose in order to maintain a more stable connection and prevent the drops and battery loss that constantly switching can cause. By all accounts I've read, the DNA does this as well, so it seems to be something that is being done more often with Verizon's devices. I would expect the Note 2 to handle things in a similar way, and I would expect it to have reception that is similar to the Galaxy S3. Some people have reported reception issues with the S3, but I haven't experienced any. Reception hasn't been as constantly strong as Motorola devices typically are, but I've found the S3 to be on par with HTC devices, which is a vast improvement from previous Samsung devices (Galaxy Nexus, Stratosphere, Charge, Fascinate, etc). I was worried about the reception on the S3 before receiving it, but after using it since July, I'm not the least bit worried about the reception on the Note 2.
I thought for sure I had double checked I was getting the correct item but it is very possible I could have made a mistake.