One key requirement for fountain (rateless) coding schemes is to achieve a high intermediate symbol recovery rate. Recent coding schemes have incorporated the use of a feedback channel to improve intermediate performance of traditional rateless codes; however, these codes with feedback are designed based on uniformly at random selection of input symbols. In this paper, on the other hand, we develop feedback-based fountain codes with dynamically-adjusted nonuniform symbol selection distributions, and show that this characteristic can enhance the intermediate decoding rate. We provide an analysis of our codes, including bounds on computational complexity and failure probability for a maximum likelihood decoder; the latter are tighter than bounds known for classical rateless codes. Through numerical simulations, we also show that feedback information paired with a nonuniform selection distribution can highly improve the symbol recovery rate, and that the amount of feedback sent can be tuned to the specific transmission properties of a given feedback channel.