how to statistically analyse accuracy data (psycholinguistics)
Sounds interesting!
Transferring counts into percentages definitely loses some information: 1/2 is not the same evidence as 100/200, but the difference is lost as a percentage.

ANOVA is pretty popular, but as you note in the question it can have problems.
Because you have the classic linguistics/psych use-case of non-independent data (ie grouped by participant), you probably want to jump on the mixed effects model bandwagon if you possibly can. I hear it's all the rage.

Some justification/motivation here:

Jaeger, T. F. (2008). Categorical data analysis: Away from ANOVAs (transformation or not) and towards logit mixed models. Journal of memory and language, 59(4), 434-446.

And R package to do the heavy lifting for you here:

https://cran.r-project.org/web/packages/glmm/vignettes/intro.pdf

Sounds like you have the binomial-input case.