On the left side of the character screen is the list of all your bridge officers. These are available bridge officer slots. If you do not have one that says Empty then you cannot commission a new one. How many stations your ship has doesn't matter if you cannot commission a new officer.
I have just got a Costellation class ship which has places for five officers. I already had four so bought an extra one. However I do not get an option to use her.
The extra slot is for engineering - so the extra officer was an engineer.
Has anyone any idea how you feel the vacant slots?
Are you mixing up consoles and BOffs?
When you look at the ship description in the shipyard, it shows icons for BOffs and Consoles. The console icons are colored versions of the little icons from inside the ToS arrows (tall red star = Tac, interlocked cblue ircles = Sci, gold swirl = engineering).
BOffs use the colored TNG combadges with the rank insigna beneath them. A red combadge with one pip under it is a tactical ensign BOff slot, meaning the BOff assigned to this slot needs to be tactical and can only use their ensign (first) power, while a 3-full-pip blue combadge is a Commander Science officer that can use all 4 of their powers.
From what I remember, no ship has 5 BOff slots until the RA (Level 40) ships, and by then you'll have like 8 BOff slots even as a silver...