Not trying to be argumentative but I applied to VCU the day before the deadline and got in so I think somebody exaggerated to you, but I do agree with you that when you apply definitely matters! I think a lot of it depends on how you are as an applicant and what the school wants but places def fill up early! I got into a few top ranked programs and waitlisted/denied at lower programs that I applied to later in the cycle (not saying that's why I got denied but one school flat out told me my app was great but they had no room for me and that was in Feb).
A lot of schools loved my work experience with those with disabilities. Another school told me that they liked how I conveyed my knowledge of the field through my personal statement so maybe try that. Good luck