What if I may not have research experience for math PhD application?
Firstly, I should mention that I am a 3rd year math PhD student at Brown. So I have a hint of knowledge, but not as much as a professor who has been to more institutions, and certainly not as much as someone who has been  on an admissions committee. 

Firstly, many starting PhD mathematicians have no real research experience. The reason is simply that research mathematics is so far from most undergraduate mathematics. This is also why doing research is so hard without a supervisor (for that matter, it's hard with a supervisor). Many candidates have done REUs, which look generically good but often don't usually result in a publication or anything.

What I recommend to you is that you:


Identify what you're interested in and pursue it. You say you maybe like applied analysis or PDE. Maybe see what some people are doing, read some of their stuff, backtrack if you need to. In this sense, you can just do research without waiting for someone else's permission. I'd like to mention that there is a polymath project you might be interested in, Polymath 7 on the Hot Spots Conjecture (here is a link to the most recent progress page). It's slowed down, but they've done some interesting things and the polymaths are generally good about leaving a clean trail of breadcrumbs.
Ask about showing research potential and getting letters of reference. Instead of worrying about what's the absolute best thing, I think you should worry about doing the best that you can do. You've tried to set up some research with professors and that fell through. You haven't done math research before, and you're probably not going to get far in the next 4 months. In particular, it's unlikely that you're going to come up with a result great enough to inspire someone you haven't worked with to write you a letter of reference. This is all to say that you should get your letters of reference from those best able to recommend you. If these are lecturers, then so be it. But I hope that you've been attentive, proactive, and inspired in those classes. For that matter, if you're inspired, you might be able to ask your dynamical systems professor for an interesting project - approachable projects exist. It might not be new research, but at least it would be something you, and others, could talk about. As a final note, you should talk to your math professors and ask them for advice - I suspect that at least one of them will be able to say something, and they're more familiar with your situation.
Have you done any research in your 'some engineering?' Conceivably, a great letter of reference could come from your advisor/a great teacher/I don't know these details. You should talk to them about this, and get their advice.
Determine what schools you're interested in applying to (apply to many), and their requirements. Brown's Applied Math department does not require algebra. But they do (essentially if not officially) require the math GRE, which does require algebra (though for less than 1/4 of the test). On the other hand, Brown's Pure Math department (the one I'm in) has a severe algebra requirement. Interestingly, PDE is done in both, though in different directions. Many schools require algebra, or require it in the sense that they will have qualifying exams and one of those is in algebra. My undergrad, which I'd say has an applied bent, had quals in algebra and both real and complex analysis. So you could consider taking algebra if you needed to or wanted to, based on looking into what universities you want to apply.

However, algebra II without algebra I sounds incredibly perilous, especially as the fall semester is upon us already, or is about to be. Perhaps possible with a lot of pre-work. If you are going to attempt to hop into algebra II, I recommend getting a copy of Dummit and Foote's algebra book (clear exposition at a low level).

Finally, plan on taking the math subject GRE. It's only offered 3 times each year, and you've missed the summer one. You should take it both times this fall, and those signups might already be occurring.