Structural Equation Modeling - Chi-Square confusion

#1
Hi,

I ran structural equation modeling in lavaan. I've run ML, MLM and MLR. The chi-square p-values are always 0.000. The data is quite nonnomal and has skewness and kurtosis.
In one case the data is 250. In another research the data is 164 respondents.

However, my performance fit indicators like GFI, CFI, NFI, RNI etc. are favorable.
In one model, of the 9 paths, 1 is not significant and in the other of the 5 paths, 2 are not significant.

Can some one guide me in this regard? What does this mean? Is Chi-Square giving a Type 1 error or are the indicators wrong?