It's actually easier to prove the stronger statement that there are infinitely many double cosets $H\backslash F/K$. 

First, note that if $F'$ is a subgroup of finite index in $F$, with $H'$ and $K'$ its intersections with $H$ and $K$ respectively, then the natural map $H'\backslash F'/K'\to H\backslash F/K$ is finite-to-one. So we may pass to finite-index subgroups.

Therefore, by Marshall Hall's theorem, we may take $H$ and $K$ to be free factors. But now the result is easy. Indeed, if $a$ is a generator not in $H$ and $b$ is a generator not in $K$, then $Ha^mb^nK$ are all distinct.