If $p$ divides $a+b$ is it true that $p$ divides $a$ and $b$? My hunch is that this is true. Consider the prime decomposition of $a$ and $b$, then $p$ cannot divide $a+b$ if it does not appear in the decomposition of $a$ and $b$, so it must divide both numbers. Is my sketch correct?

Your assertion is not correct.

You can say that if $p $ divides $a $ then it must divide $b$, because $p$ divides $a+b $ and $a $ so it divides $a+b-a=b $, in a similar way if $p $ divides $b $ then it must divide $a $.

Your assertion is wrong as you can take $p=2$ and $a$, $b $ any two odd numbers, then $p $ divides $a+b $ but it does not divide $a $ nor $b $.