Is there a way to prevent this"weeping"?
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I've heard that including a spoonful of cornstarch in the filling can help prevent that, but I've never actually used cornstarch in a pumpkin pie myself (I have a weird affection for soggy pumpkin pie crust! Though not in any other kind of pie.).
Over baking may be part of the problem. The eggs in the filling can become heated to a point where the proteins start squeezing the water out (think overcooked watery scrambled eggs at the hotel breakfast buffet). Adding starch or flour, as stated above can reduce the risk of this happening (my pumpkin pie recipe includes a tablespoon of AP flour). Most recipes, including the one I use, recommend removing the pie when the filling is still a bit giggly in the center. The residual heat from the outside of the pie will complete the cooking process for the center of the pie as it rests on the counter.
I agree, the custard is either being baked for too long (it should still wiggle in the center, but it shouldn't slosh at all when you take it out of the oven) or it is baking at too high a temperature (get an oven thermometer and check the part of the oven where your pies are, non-convection ovens can have hot and cool spots as well as having temperature sensors that aren't calibrated correctly). Also allow pies to cool completely before refrigeration to prevent condensation; custard pies must be refrigerated, weepiness can also be caused by bacteria.
I've never had one weep so much that the crust got soggy though, usually it's just a few sticky beads on top of the custard when I over-bake to the point of pumpkiny abysses. Maybe it's the recipe? I use the "Ultimate Pumpkin Pie" recipe from Greene on Greens which does not use a starch in the filling.