Here's an easy way to see if it's the shifter itself...
Take the cable out of the shifter...Click it all the way down to the highest gear...shift one gear at a time back to low and count how many clicks you hear, it should be 8, if it isn't your shifter is broken...
Easy way to check limits...
Without a cable in your rear derailleur, using your hand, push it towards the low cog, it should go all the way, if not, back out the "L" screw until it does...
Easy way to check cable tension...
Cable should be taught, not tight, click one gear on the shifter, the derailleur should move one gear without hesitation...Adjust barrel on derailleur until this happens...If it wants to jump 2 gears, the cable is too tight...
Otherwise, drop it off at a shop and let a mechanic figure it out because it's not something you might be able to fix, like a bent derailleur hanger...
-Jim.