I'm confused how it seems like it's in 6th when you're still in 3rd. If you can't shift into any gear above third you have a problem for sure. I may be wrong but I thought there was a tach function built into your speedo display. Check your owner's manual for it.
Maybe you can get a friend with more experience to ride the bike and tell you what's going on.

 

maybe your lower gears are not working and you are starting off in a higher gear?

 

Do you have any way to lift the rear wheel off the ground? If so, with the motor off, see if you can engage all six gears. Sometimes you need to rotate the rear wheel slightly, but it should easily engage each gear. When you say you can't get into 4th... Does the lever move, does it act like it changes but you aren't aware of an RPM difference... Like, what actually happens? Does neutral exist in the right spot, between 1st and 2nd, or is it possible that you're starting in 2nd and reaching 4th?

My 5-speed turns 3500 at 80... I would think I could hit 80 in 3rd, maybe around 4500 rpm. My engine runs fine in that range, but I would be very aware that it was spinning quickly...

 

We're thinking along the same lines, at the same time!

 

push the trip button, till it shows rpm's, and it will show what gear its in when the clutch is out.

 

Great minds !
Maybe we will find out of we are right / wrong at some point.

 

Is the shifter linkage sloppy? are you dhifting to "every other " gear? 1-3-5 for example?

 

1 down, 5 up.... are you starting in 2nd and breezing past one of the other gears?

 

"No redlining, and really smooth",

" Honestly it’s hard to tell without a tach. "

how do you know about the red line if you havenae a tach

it may do the speed in 3rd, but the engine will be shouting to you