Sorry for not being clear. By door open alarm I meant the bing bing bing noises and dash indicators telling you your door is open. Door open alarm. Or alert. Or notification.
You said you can turn on the dome light when it is running. Is that only if you manually reach up and push the button, or does opening either door while the engine is running turn on the light? Does opening either door while the engine is running activate the door open alarm/alert/notification?
The switch is probably the most common problem, but I don't think that would prevent the door open alarm/alert/notification from triggering. So if it's the switch on the stalk going bad, you might actually have two problems. Knowing if the problem is just one door, or both, can help point to possible causes.