HerHusband wrote:
....

Unfortunately, for a variety of reasons I do not want storm doors on the
doors that are leaking. Especially for the garage and back doors where
we're often hauling large objects in and out. My in-laws just splurged on
a decorative door and don't want to cover it with a storm door.

Well, you can solve the problem or replace the sills over and over, your
choice...

There are full-exposure s-doors that won't hide the face of the inner
door if that is a requirement.

W/O pictures including installation details it's hard to make any
specific conjecture but I'd say the likely culprit is there is no slope
outward so water stays where it lands rather than drains. As well,
capillary action may be pulling some under the sill if there is any
small crack at all.

Depending on where and how they were caulked, it (the caulk) may be
server more as a drainage dam than as a sealant.

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