MinJune

Hi, I'm an undergraduate majoring in pure math. I did not have any luck with PhD admission for math this year, but I was accepted to the above two programs. My background: I think my undergraduate institution is very reputable; its math department ranks around 6~10 in the US, and overall is very well recognized. I also have taken all their PhD level analysis core courses (totaling 4) and many pure math senior level courses. My failure in the PhD admission cycle is due to the lack of internship or research experience due to my personal condition. So, in a way, I think pursuing masters degree is a reasonable choice right now. I want to keep pursuing PhD program, possibly applied math this time. I have very little knowledge in it, but I'm interested in the subject. Do you have any advice for me, in order to become stronger candidate for the top applied math programs in US? Is it acceptable to discuss this kind of thing with professor (maybe it can be seen as too mercenary?)? Further, can you shed some light on which program is better? UWashington or UChicago?