graduate school - Few Months Left to Prepare for Pure Math PhD Application

I'm hoping that I can get some general advice on strengthening a graduate application in the last few months through introducing myself to some faculty in my current program.

I have recently landed on the decision that I want to apply to a pure Mathematics PhD program this round (Winter 2018/2019) and I'm becoming more and more aware that I am not going to be very competitive. I have average Math subject GRE scores, and no rec letters to speak of.

The reason I haven't cultivated any rec letters is that I am currently a graduate student in an Applied Math department, working toward a PhD, and until recently had never expected I would be re-applying. I have been working in an area that has -- over the past couple years -- become less and less interesting to me, to the point where I cannot see myself happily working in this field. In particular, the program is not mathematically rigorous, and the work is more engineering without even requiring us students to grasp the fundamentals of advanced math (the whole point of studying math, for me).

I have decent grades in my courses so far, and have obtained the requirements for a Masters in Applied Math. I have a pre-print of a paper in the pipe for publication, with two well-established co-authors. The lack of recommendations scares me, though.

A friend suggested that I try to "throw myself into the math" by basically approaching several different faculty members, across various departments (physics, math, etc.), and becoming known to them through office hours, auditing courses, small projects, etc. In the end (by Dec), I should hope to get some nice letters.

I am hesitant to do this because I don't have a solid foundation in the advanced topics (analysis, algebra), as I haven't had to take these courses. I am currently and have audited several intro-level pure math courses, and couldn't be more eager to learn, but I don't feel prepared to approach professors, much less ask them to spend their precious time on me.

Another friend suggested I create a blog where I do periodical write-ups of some interesting math topics. I might just do this, even if only for fun. I'm not sure how this would even end up factoring into an application, however.

Given my circumstance, can you offer any advice on how to make the most of the remaining few months before applications are due? Is forcing my way into professors' lives for a few months a reasonable way, at this point, to try for recommendation letters? Or would I be better of trying to pursue some small projects with my current advisor that I can neatly write up and post online?

Thanks much for any advice!