Replaced battery, car died again.

Im sorry to hear about your car but here's a video that might help you figure out what's wrong with her. If anything try to get under the car and look around for any damage caused by the rain. (Link) https://youtu.be/Yehz2gUnNe0

So I don't think that's the answer. I tried starting her up this morning and she started ok, but I turned on the radio and lights and the radio shut off within five minutes like before while idling and then I shut her down.

The radio may be a red herring. This is a tough diagnosis without seeing the car. Maybe let things dry off (hood open); start car, put AC on to operate rad fans and leave hood open to dry off ignition, etc. Then drive normally. If motor is happy but mismatch of RPMs and speed are evident then check trans function. Any CELs? AT light blinking? I'm hopig you haven't coincidentally lost your AT.... There's a decent chance you got a batteryy that's shorted internally or won't hold a charge, so don't discount trying another one if starting remains weak. I need more coffee for this one....

I have today and tomorrow off, but its raining today and I'm parked on the street. So tomorrow I'm going to replace the battery again see if that helps. Also I think the connector cables might be shot, they are highly corroded and rusted, I cleaned them but it might not have been enough.

The thin brass battery positive "half-rings" get eroded over time, reducing current carrying capacity. Best you head to any store and get a $2 connector, clean up the cable end, and screw 'er together. Clean, and repeat, if necessary, with the negative.

I am having the same problem now, on a 2093 Forester. How did you end up fixing it?

Clean battery terminals and clamps thoroughly and jump-start. If repeats check alternator voltage. If under 13v replace.