Sunbird 94 stalls / starts with a delay

I have a 94 sunbird past the 300,000 km. A week ago I started to have problems. When I turned the key, sometimes it would take 1 second or two before the engine would engage. I would turn fine after that. and a few days ago, at idle the car would idle super fast and starts shaking, when the accelerator is applied , it would stall. Agter 4-5 stall in a row it did not make this problem again. Also when it is below 20degrees. the car will have a hard time starting and sometimes even after removing the key, the starter was still trying to start the engine. whick OBVIOUSLY drained my battery. I changed my battery a mnth to a

thanks for any help ken