Fixed effects versus standard errors

#1
I have experimental data from a correspondence test in the rental housing market.
On which ground should I choose between:
1) F.E. for the region where the apt is and cluster s.e. on the day I sent the message
2) cluster s.e. on region and F.E. on day when the message was sent (or, perhaps, better week end than day)
More in general
On what ground can I choose whether to control for one variable or to cluster the s.e. on that variable?
The first option changes the estimate of the coeffs while the second changes the estimate of the s.e.'s....both of them should be used when I suspect there is something which is unobservable and is shared amongst some observations.