No you definitely get paid for every round you win.
Have a look at the FA's prize fund breakdown Emirates FA Cup prize fund
"Amounts are per club in each round. Numbers in brackets represent the number of clubs receiving the respective payment per round"
"If the FA Cup winners finish inside the top four thereby qualifying for the UEFA Champions League — or fifth, which qualifies them for the Europa League — then the Europa League qualifying spot offered to the FA Cup winner will be reassigned to the next-highest-ranked Premier League team."