You say "yes", but then you contradict that with "within a few weeks." Details and exactness are important.
You have multiple different problems, most of them known with the LS. Search the forums here.
Door ajar - The switch in one of the door latch assemblies is failing. You have to replace the latch assembly. eBay/junk yards are the only sources left for most of the latches.
Front windows but not back windows is odd. Usually, it would be the key lock/unlock switch on the driver's exterior key lock shorting out, but that would open/close the rear windows too. I think that this would have to be the master window switch assembly in the driver's door armrest.
It sounds like your DCCV is clogged, or the cooling system is air-locked due to micro-cracks in the plastic. If the cooling system plastic hasn't been replaced in the last seven to ten years, there it is.
It also sounds like you are low or out of refrigerant or the DCCV/clutch fuse is blown in the under the hood fuse box. If it is, then your DCCV is shorting out and must be replaced.