By the 1860's why did Americans no longer seek to resolve their political dispute through compromises?

There are two reasons why Americans stopped trying to
reach compromises to resolve their political disputes.  (I assume you are talking about
the issue of slavery.)

First, they had tried various
compromises and none of the compromises managed to satisfy both sides.  Both the
Missouri Compromise and the Compromise of 1850 helped for a while, but both eventually
broke down.

Second, and perhaps more importantly, the two
sides no longer trusted one another and were not really all that interested in trying to
get along.  The events of the last two decades or so had hardened the two sections'
attitudes towards one another.  Compromise relies on the two sides having at least
somewhat positive attitudes towards one another.  The North and South no longer had this
sort of attitude.