Lafayette traffic can be bad. Your best route thru Lafayette is as Laman advised. The Ambassador Caffery and Johnston St (US167) is a major shopping area so if you can time it so you are not there on a weekend afternoon you will be better off.
Going East to Alabama you really only have two choices - thru Baton Rouge or New Orleans. The choke point in BR is that I-10 narrows to 1 lane coming off the Mississippi River bridge. Best bet is time so you pass on Sunday morning.
On either I-10 or US90 there are rough sections.