In general, how does one resolve two competing theories in Psychology? In looking at a post on twitter from a somewhat well-known psychologist Tal Yarkoni, he references a paper which lists multiple sets of competing theories in Psychology that have not been resolved.  

This is interesting to me and related to my question on selecting "the most well supported" among competing theories of generalized anxiety disorder (link).  But, I could see this being related in other areas, such as procrastination research where psychologists with different focuses explain the phenomena differently (as elaborated in this video.)

As an aspiring research scientist, I am curious what it takes today to resolve competing theories in Psychological science?  I read that it may take a joint research paper between the authors of the differing theories with conditions of what defines sufficient evidence, but certainly there must be another way, because some authors may be deceased, unavailable, etc.

Moreover, why might these disputations, it seems like there are 13 of them (26 theories in total), some dating back to the 50s, still be unresolved today?

I think this question should be germane to the topic, because in a meta post here, the moderator agrees that meta-science posts are fair questions for this site.
Almost every pair of theories on that list sets up a false dichotomy.

It is not an issue of one theory over another; at issue is how much different explanations apply to different circumstances. There is at least some evidence for each possibility that is not explained away by the other.

Resolving these dichotomies in favor of one theory or the other is probably not a worthwhile pursuit.

However, many papers are published that are greatly informative on some of these issues, and in fact if you had scrolled down just a bit on your own link or followed up a bit, you would find that the original author of the table presented in that tweet has an updated list (though several years old now) that mentions several such papers: https://faculty.washington.edu/agg/pdf/PPS_2012_supplement.Unresolved_controversies.pdf

I especially like this footnote from that link, which I think strongly supports my answer (emphasis mine): 


  b This controversy is treated as unresolved by some researchers, but the recent article by Smith
  and Grossman (2008) has argued, using brain imaging evidence, for “multiple systems of category
  learning”, which suggests a controversy resolution in the form treating multiple theories of the nature of
  category representations as valid in distinguishable domains of application.


So to answer your question:


  In general, how does one resolve two competing theories in Psychology?


In general, one resolves the existence of competing theories by testing cases where one theory or the other better explains an experimental result and attempts to resolve the conflict by finding mediating factors that prefer one theory over the other (or some entirely separate theory that may explain all or part of the data).

This is usually an incremental, iterative process, not something that is resolved in one key work.

References



Greenwald, A. G. (2012). There is nothing so theoretical as a good method. Perspectives on Psychological Science, 7(2), 99-108.