If you want to assess all funds together, i.e. see if the seasonality is prevalent for all funds as a group, then follow Dimitriy's advice. If you would also like to inspect some funds separately, you could still use the regression you have. Just recall that by testing $n$ funds one by one, you would end up rejecting the null hypothesis of no seasonality $0.05 \times n$ times even when none of the funds were seasonal. Refer to the literature on multiple testing corrections then.
Testing for autocorrelation
Durbin-Watson test targets only first-order autocorrelation. For monthly returns on funds this could probably be sufficient; you could consult financial theory on whether higher-order autocorrelations could be expected. If checking higher-order autocorrelations were also of interest, you could use Breusch-Godfrey or Ljung-Box tests.
You could also see if the model residuals have autoregressive conditional heteroskedasticity by using ARCH-LM test (ibid.). By neglecting heteroskedasticity when present, you could lose some power of your tests.