PCSE vs FGLS method for analyzing panel data


Some guidance on the below inquiry would be much appreciated as I am having a hard time finding someone who is faimiliar with the PCSE and FGLS methodology. Thank you very much for your help.

I have been using the PCSE regression method to analyze my time-series cross-sectional data; however of the three panel error assumptions (groupwise heteroskedasticity, contemporaneous correlation, and serial correlation), the PCSE method only corrects for the first two. I tested for serial correlation using the Wooldridge xtserial test. Thus the problem of serially correlated errors needs to be addressed before running the PCSE model or it can be addressed by specifying PCSE-AR1 in Stata (indicating presence of autocorrelation) and a Prais-Winsten regression will then be used to correct for the serial correlation. If I try to correct serial correlation by including a lagged dependent variable or by specifying it in the model function (corrAR1) my results change completely and most relationships are no longer significant. My guess is that this is an over-correction and really isn't necessary; however I cannot justify this based on my level of stats knowledge or from related literature.
Complicating matters is that I have also tried running the regression using the FGLS method which supposedly accounts for all three of the panel error assumptions; however based on literature especially by Beck and Katz, this method seems to only be appropriate when T is at least three times as large as N (for my data T is twice as large as N). Also, most recent studies using panel data prefer to employ the PCSE method. Oddly though, the FGLS and PCSE methods produce identical results in Stata when autocorrelation is not specified, which from my perspective implies that PCSE method is also doing some sort of correction for serial correlation making a further correction unnecessary. The problem comes in explaining the methodology of the study. Since PCSE is the preferred methodology I do not want to include a lagged dependent variable to correct for serial correlation or specify AR1 in the function because this leaves me with basically no significant findings to discuss. The results from just running PCSE or FGLS on the other hand produce several significant coefficients that we can explore. Is there a way to justify not specifying AR1 or to justify using FGLS? or perhaps another way to correct for serial correlation that may not have a huge impact on the results?