Options for you: McNemarNP, $t$ test (with variants)P, WilcoxonNP, sign testNP, FriedmanNP
Five statistical tests are compared primarily on the type I error produced. Emphasis mine.
Two widely used statistical tests are shown to have high probability of type I error in certain situations and should never be used: a test for the difference of two proportions and a paired-differences $t$ test based on taking several random train-test splits. A third test, a paired differences $t$ test based on $10$-fold cross-validation, exhibits somewhat elevated probability of type I error.
In your case, this shouldn't matter as a $t$ test assumes normality to a certain extent.
The cross-validated $t$ test is the most powerful. The $5×2$ cv test is shown to be slightly more powerful than McNemar’s test. The choice of the best test is determined by the computational cost of running the learning algorithm. For algorithms that can be executed only once, McNemar’s test is the only test with acceptable type I error. For algorithms that can be executed $10$ times, the $5×2$ cv test is recommended, because it
is slightly more powerful and because it directly measures variation due to the choice of training set.
Note that McNemar's test is non-parametric, similar to goodness-of-fit that uses a $\chi^2$-distribution.
This paper is more interesting as it considers some alternative, non-parametric approaches, such as the Wilcoxon signed-rank test.
When the assumptions of the paired $t$-test are met, the Wilcoxon signed-ranks test is less powerful than the paired $t$-test. On the other hand, when the assumptions are violated, the Wilcoxon test can be even more powerful than the $t$-test.
Another non-parametric test is the Friedman test. It is similar to ANOVA, and still uses ranking as part of the expression for the test statistic.
A simpler method is to use the sign test, and the larger the number of data the closer the equivalence of this test to the $z$-test. However, the cost of the simplicity is highlighted below.
This test does not assume any commensurability of scores or differences nor does it assume normal distributions and is thus applicable to any data (as long as the observations, i.e. the data sets, are independent). On the other hand, it is much weaker than the Wilcoxon signed-ranks test.
Overall, the non-parametric tests, namely the Wilcoxon and Friedman test are suitable for our problems. They are appropriate since they assume some, but limited commensurability. They are safer than parametric tests since they do not assume normal distributions or homogeneity of variance. As such, they can be applied to classification accuracies, error ratios or any other measure for evaluation of classifiers, including even model sizes and computation times. Empirical results suggest that they are also stronger than the other tests studied.
 Dietterich, T. G. (1998). Approximate Statistical Tests for Comparing Supervised
Classification Learning Algorithms. Neural Computation. 10:1895-1923.
 Demšar, J. (2006). Statistical Comparisons of Classifiers over Multiple Data Sets. Journal of Machine Learning Research. 7:1-30.