if( $@ =~ /some error stuff/ ) {
    warn "error stuff occurred";
} 
else { 
    warn "some other message" ;
}
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As I said I usually just do

if($@) { Oops; }
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but now you mentioned it and I remember one case when I have to examine $@ content deeper. But usually I'm avoiding this, particularly because I don't see the way to do it clean.

I feel like Perl exceptions are kind of like Perl OO in general.

I feel that too. The problem is that $@ may contain any object, there's no standard interface to it. If there will be a module that change this situation, that would be great.