I'm trying to get 6 different crew members on the ship but my crew is full; should I kill a current crew member to ensure space to take on new crew, or will I be given the option to swap if a new potential crew member offers their services?
Yes, you will be able to gain new crew members. When you get a crew from an event, e.g., an interface will appear at the end of the event showing you every new crew member + the new guy (who will helpfully be flagged as "new"). You then have to chose to dismiss a member since you are over the limit. If you see crew for sale in a store, you can go into your ship screen and dismiss a member there so that you can buy a new one.
In the comments, ChargingPun confirms that when you are on the "crew dismissal" screen, you can still mouse over crew members to see their skills.
Also please note that as of patch 1.5.4/the recent add on, potential crew you encounter can come with skills which have gained experience. It is possible to get a crew with max [insert skill]. This is tremendously useful, as it both allows one to more easily make up for later-sector mistakes as well as maybe find someone with decent shields skill (heh).
When you have a full complement, and a new crew tries to join, you will be taken to the crew selection menu where you have the option to fire a crew. You can choose to fire the new crew or an existing member.
To supplement Peacedog's answer, the following screen shows up when you try to recruit new crew members, even if your roster is full:
In this case, you can see my existing eight crew members, and in the bottom right, the new Human, Maria, who would like to join my crew. Since I have already reached the maximum number of crew members, I can either
dismiss an existing crew member, or
dismiss the new recruit.
If you are having trouble deciding who to get rid of, you can hover over a crew member (and I'm kicking myself for not including it in the screenshot) to see their current skill levels. Keep in mind that new recruits in FTL:AE can come with pre-trained skills in later sectors.