NBEEBE,
If you're using AWS D1.1 (Structural Welding Code), Refer to Sections 2.4.5.3 and 2.4.5.4
The effective area of the plug or slot weld is generally equal to the area of the hole or the
slot being welded. This is applicable for the load putting the weld in shear.
Awe CANNOT be area of the slot X depth. That would result in a VOLUME in in3 / mm3, not area which will
give you incorrect results.
As mentioned above, if the load is perpendicular to the plane of the weld, then the weld should be in tension. I'm not aware of a Code provision for tension on a plug or slot weld. What was mentioned seems to make sense. However, instead of perimeter X thickness, perhaps a fillet weld around the perimeter of the hole or slot and using the effective throat would be more appropriate. The remaining area would then just be filled with weld. But I would confirm this is in compliance to your Code regarding tension on a plug weld. I'm not sure what qualifications may be required for such welds or tension strength of such welds.
So to the root of your question, the depth of the plug / slot weld does not go into the calculation for effective area or weld shear strength. However, the required depth of welding must be provided even though it is not directly a factor in the weld strength.
One final note; in addition to checking the weld strength, there should be other limit states related to the base metal(s) that should be considered such as bearing, tear-out, block-shear, etc. similar to what would need to be checked for a bolted connection.