Can I still submit my work to a journal if a very similar paper was just posted on arXiv? I am from an engineering field (EE, specifically signal processing), and I have been working on a paper since March 2014.

Back then, I spent a significant amount of time reviewing the literature and didn't find any existing work that discussed my topic. I actually finished the bulk of the paper several months ago (Jan 2015), but since then I have been editing and reviewing it with my coauthors before we planned to submit to journals.

I recently, however, found a similar paper that was "published" or posted on arXiv in April 2015. I believe that it is extremely similar (~80% of the ideas). I am not sure how much of an issue this is, as I have heard that arXiv is not a "legitimate" domain (in my field, almost everyone just uses the IEEE searching engine). 

My original plan was to submit my work to journals soon. Do you think I can still submit my paper to a journal? How should I acknowledge of otherwise handle the existence of this similar arXiv publication?
Sure, you can go ahead and submit your paper. Of course you are obligated to discuss the arXiv paper, and its relation to your work -- but it sounds like you can honestly describe it as independent work. It would be a good idea to also make your paper public at this point (perhaps by posting a preprint on arXiv).