How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit?

$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 60  \qquad\textbf{(C)}\ 66 \qquad\textbf{(D)}\ 68 \qquad\textbf{(E)}\ 70$
We use a casework approach to solve the problem. These three digit numbers are of the form $\overline{xyx}$.($\overline{abc}$ denotes the number $100a+10b+c$).  We see that $x\neq 0$ and $x\neq 5$, as $x=0$ does not yield a three-digit integer and $x=5$ yields a number divisible by 5.
The second condition is that the sum $2x+y<20$.  When $x$ is $1$, $2$, $3$, or $4$, $y$ can be any digit from $0$ to $9$, as $2x<10$.  This yields $10(4) = 40$ numbers.  
When $x=6$, we see that $12+y<20$ so $y<8$.  This yields $8$ more numbers.  
When $x=7$,  $14+y<20$ so $y<6$.  This yields $6$ more numbers.  
When $x=8$, $16+y<20$ so $y<4$.  This yields $4$ more numbers.  
When $x=9$, $18+y<20$ so $y<2$.  This yields $2$ more numbers.  
Summing, we get $40 + 8 + 6 + 4 + 2 = \boxed{\textbf{(B) }60}$