Is there a verb base-form with two semantics having different conjugations?

Solution 1:

The past tense of hang has two different conjugations.

When referring to the method of execution:

He was hanged at dawn.

When referring to any other purpose:

The picture was hung on the east wall of the bedroom.

Solution 2:

It’s only in writing, but supposedly “payed” is used instead of “paid” for certain meanings of the verb pay. See "Paid" vs "payed"

Furthermore, there is a hypothesis based on theoretical considerations that the verb fly out as used in baseball ought to have the past tense form “flied out” instead of “flew out”, but the empirical support for this hypothesis seems to actually be fairly modest—speakers (or writers) do in fact use “flew out” even in the context of baseball. See the Language Log post “Flew vs. Flied”, by Mark Liberman (2012).