From my online research, I have found American Airlines to be the most constant for excellent Senior flights deals for those 65 and older. Recently, my mother was unable to take her return flight from across the states. My sister and I worried that they would charge her extensively to switch flights so we were glad we had purchased the traveler's insurance (something I've always done since I got the flu the night the same night we rented a car for the next day and had to cancel; that was $90 down the drain) but the insurance required a doctor's note. To our pleasant surprise, American Airlines said all she had to do was pay the difference between her current flight and the one she was switching to (in her case $5 difference). They have excellent customer service and said they would allow a refund or change of flights even if she had changed her mind. In order to get a Senior flights in the drop down menu where it is asking the number of Adults traveling, click the highlighted area below that says "Seniors, children and infants." Fill out the more detailed form and select 1 passenger in the "seniors" drop down menu, then just make sure that is the setting you are using when you search. Click HERE to view their Customer Service Plan.
American Airlines no longer is participating with many online flight search companies like Orbitz, Expedia, and Hotwire so search directly on their site and you will get there best deals. They have a lowest fare guarantee HERE and no online booking fees.