I have a two by two design. Ideally I would run the whole thing within-subjects (all participants get all conditions) but the procedure is too long. I could run one factor between-subjects (each participants gets one condition) and one factor within-subjects(each participant gets both conditions), but then I have more power to see the effect for one factor vs. the other. I was thinking about running it such that one participant gets factor A within and another participant gets factor B within. That way I have to recruit fewer participants and I can see within subject and between subject effects for each factor. But I'm not sure how this would affect the analyses. Advice would be appreciated.
Don't try to come up with a complicated analysis to deal with this, just do within and between analyses separately on the relevant subsets of your data. That way you will not compromise the interpretability of the outcome and you will be able to make sure that the result is both sensible and robust.
Depending on the nature of your data you could potentially combine the outcomes of the two arms of the experiment. For example, if you can generate likelihood functions for each arm then the combined function is simply their product.