The fields of 
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- oo_.PosteriorTheoreticalMoments.dsge
denote the moments you are looking at. The next field provides the statistic for that moment considered. 
1. 
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- oo_.PosteriorTheoreticalMoments.dsge.covariance.Mean.y.y
 should be the mean covariance between y and y, i.e. the variance. What you wanted to look at is the variance of the covariance over the MCMC draws.
2. The autocorrelation of y, starting from lag 1, should be in 
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- oo_.PosteriorTheoreticalMoments.dsge.correlation.Mean.y.y
 
If you want contemporaneous correlations, you need to use the unstable version with the
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- contemporaneous_correlation
option and then look at e.g. 
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- oo_.PosteriorTheoreticalMoments.dsge.contemporeaneous_correlation.Mean.y.c
 
 to get the correlation between y and c