That 1st gear shift is the "limp-home" mode that early 4-speed transaxles are known for. Keep in mind that it may not be the trans itself...those cars have a seperate computer for the transaxle, along with a few relays that are right by the computer (passenger side under the hood, bolted onto the inner finderwell I think).
I had a 95 with that setup and the only problem it gave me was the timing belt broke while I was on a local cruise. But other than that, they are very good engines and can be modded a little bit (upper intake plenum can be cleaned up considerably). The rear windows often fail, and have an intricate pulley system that you will have to watch out for when/if you replace the motors. And there are ways to adjust the top to make it fit snug. Most people don't like the LeBaron, but I think it'* a great car and really fun to drive.
Welcome, downwardly mobile yuppies, to the dawn of the Age of Nefarious.
1993 SSEi - FrankenJinx - dead and gone.
1994 SSE (136K strong) - 180* t-stat,HD alternator, most unequipped SSE on the club, XM radio, Kenwood CD/MP3.
2006 Chevy Malibu Maxx LTZ - Custom 2.25" manifold-back exhaust w/FM-80. Probably the only Maxx with this system.