Trying to book Anaheim Fairfield Inn for 2016 (first part of march or last part/beginning of april/may) and can't find ANY dates that are available. :/
I see lots of availability at the Fairfield for the time periods you suggested. I am looking at Marriott's site. I think the only weekend that was unavailable was 3-11.
I know a lot of hotels are booked for the weekend of 3/11 because of the Natural Products Expo, but otherwise I think you are just seeing a glitch in Fairfield's system. Did you try searching for your dates individually?
The end of April/early May also have limited hotel availability at this point because Bats Day in the Parks and Spring Dapper Days are slated for Apr 29 to May 1 and on May 1, respectively. The following weekend (May 5-8) is the Tinkerbell Marathon.