Is O(|A| + |B|) the size of the union of two synchronized DFAs A and B?
Your conjecture is wrong. Consider the following languages:


$L_1$ is the language of words over $\{a,b,c\}$ such that the number of $b$s after the last $a$ is a multiple of $x$.
$L_2$ is the language of words over $\{a,b,c\}$ such that the number of $c$s after the last $a$ is a multiple of $y$.


The minimal DFA of $L_1$ contains $x$ states, and that of $L_2$ contains $y$ states. Both DFAs have $a$ as a synchronizing word. The minimal DFA for $L_1\cup L_2$ contains $xy$ states (and also has $a$ as a synchronizing word).