Spaniard"and "Frenchman" are both perfectly acceptable!
But old fashioned.... my tendency is to mostly use "national adjective" and people in the plural "French people" "Spanish people"
In the singular I mostly just use adjectives "He's French" "She's Spanish"
I would mostly rephrase sentences to avoid Frenchman or Spaniard the same way I'd rephrase to avoid using 'one' (cause it's hard to use that without sounding like a pedantic twit - I don't mind sounding like a pedantic twit when occasion demands, but I don't like to make a habit of it...)