How clean are the battery terminals? I know you replaced the battery, but did you clean off the contacts on the cables? I had an old Chevy Blazer just up and died one day. I had replaced the battery already and it just wasn't getting any juice. I took a wire brush and some water/baking soda mixture and cleaned them off and it fired right up. That doesn't really sound like your problem but it's free and should be done every once in a while anyway, so try it.

I'm still not sure what your car is doing. Does is sound like anything is spining or just locked up? Has the car overheated lately? Is the oil full and clean and is the radiator topped off? Seeya.