Had a similar issue on my 2000. Needed a new fan (yes, all 2000-2002 LSes have a hydraulic cooling fan). It was running, but the pump wasn't pumping enough fluid to spin the fan fast enough. On the freeway, fine, slow down and the temp would shoot up. As the pump and fan is an assembly, both had to be replaced. IIRC, it was quite expensive. The part is ~$400 and about 3 hours labor.
IMHO, you need a new pump assembly. I'm pretty sure it isn't a DIYer job. There are a couple of special tools required. There is also a special bleeding procedure.
IMHO, you need a new pump assembly. I'm pretty sure it isn't a DIYer job. There are a couple of special tools required. There is also a special bleeding procedure.