Why didn't I beat the avalanche? Yesterday, while skiing out of bounds on the south west canyons of Mt. Hood, I experienced a small and extremely mild quake. This, combined with the melting conditions caused an extremely small avalanche in the canyon region we where in. 
This was extremely mild in the spectrum of what is shown on TV or from what I've seen in other videos. A snow field of a bout 100 sq.m, shifted about 100 m down hill. And while mild, this was still my first ever experience in this type of situation. 
Im a fairly experienced skier and we were already traveling downhill at about 35  mph +/- (we were tracking with a racing GPS app), when the snow "gave out from under". In normal "powder" and "good" snow, it normally feel like im floating on the snow. In this situation, it felt like I was being pushed from behind, while sinking under. Maneuvering was next to impossible and the only option to accelerate was to "tuck". While initially I accelerated a little, the snow caught me and my group rather easily and rapidly. Gladly, no one was hurt, but my questing are as follows:
1) If all free falling objects accelerate at the same rate (this was on a fairly steep mountain section), why did we get "trapped" into the avalanche, when our acceleration already had 35 mph +/- accelerating it? Did the thicker snow breaking and shifting somehow create more friction between my ski and the snow?  
2) Most modern ski's today have alot more "nose curve", also known as rocker, in the front and back, in order to break through thicker snow with more ease   The skis I was riding have what most in the ski world would call "excessive rocker". Why was it so easy for snow to sink me down, when my ski's design is for 2+ ft of snow and I was forcefully trying to "float" above the collapsing snow? This avalanche was tiny in the spectrum of perimeter and mass traveled., And also only lasted about 3 seconds. 
 A: Tl;DR: It is mainly due to the skier's positioning on the ski and how being thrown off-balance by the avalanche affects it.

Concerning the first part of the first question:

1) If all free falling objects accelerate at the same rate (this was on a fairly steep mountain section), why did we get "trapped" into the avalanche, when our acceleration already had 35 mph +/- accelerating it?

there is probably a large psychological component here. You were moving across a stationary ground and suddenly that ground starts gliding, which it didn't before. So it is really the change in its movement that you felt.
There is also a physical component however: you state correctly that "all free falling objects accelerate at the same rat" and under this assumption it shouldn't make any difference that the ground started moving. But a free fall approximation is not very good for skiing, especially when you already are in motion.

Did the thicker snow breaking and shifting somehow create more friction between my ski and the snow?

The answer is of course no, on the other this question already hints at what actually happens. Skiing is a large dependent on your position and balancing on the ski, and the acceleration does too (1). Our case is different from racing, but similar concepts apply. In the "dust on crust" you describe you have to lean back slightly to float, even with your rockers. This comes at a cost of acceleration. When the avalanche starts moving you get thrown off balance a bit, leaning back even further relative to the ground. To accelerate, you get back in front first applying the pressure to the ski as before. This takes some time of course.

Why was it so easy for snow to sink me down, when my ski's design is for 2+ ft of snow and I was forcefully trying to "float" above the collapsing snow?

Same reason, due to the leaning back your ski doesn't float properly anymore.

(1) which is e.g. why you we saw Ted Ligety win almost every Giant Slalom a couple of years ago: he invented a better ski position
(2) aaaah, second best after deep powder in my opinion
