How to draw NPDA for words whose number of b's is strictly more than that of a's but strictly less than twice the amount
As you knew intuitively, you just missed the solution by a millimeter, which, however, may look like a mile away.

$S\rightarrow aabbb/aSb/aSbb$

The idea is simple. Since $ab$ and $abb$ should not be generated, what is the shortest or simplest word that can be generated?

Here is a sketch of a NPDA. When it reads the first $a$, it goes to state $q_1$. When it reads the second $a$, it goes to state $q_2$. Then every time if it reads one $a$, it will push one or two $a$'s nondeterministically onto the stack. Then when it reads the first $b$ it goes to state $q_4$.  When it reads the second $b$ it goes to state $q_5$. When it reads the third $b$ it goes to state $q_6$. Then every time if it reads one $b$, it pops out one $a$ from the stack, staying at state $q_6$. 

I will let you flesh out the details.