To English ears, it's less a hard «Я» sound as much as it is closer to «и». I think it's just that the English «y» sound of the «я» is minimised compared to words like «моя» оr «п'ять».
I'm having a difficult time pronouncing the 'надцять' part of many of these numbers. Could somebody please break down the pronunciation for me?
Which part is the trickiest?
Raise the middle of your tongue as you finish "на" and then use the blade of your tongue to pronounce the palatalized /дц/ and /т/. After that, practice getting the voicing right (дц is a bit like "ds" in "needs", with a hold, but not quite)
Note that Ukranian Д and Т are pronounced closer to you teeth than in English.