Your oil light is not a change oil light. It comes on when the sender unit sends a signal to the oil light because it detects low oil pressure. I don't know if a short could cause this. You had an intermittent oil light that is now constant. That both describes a floating blockage in the crankcase and it describes wire damage causing an electrical short.
You said that your oil dipstick is broken. Which is why I went right to advising checking your oil pumps pickup screen for blockages. While your there you can needle nose that broken dipstick out of the bottom of the dipstick tube and enable you to replace it. You don't need to remove the upper structural section above the lower oil pan to access either of these things.
If that's clear, you get to be sure that your oil pressure might not be constricted and you can begin testing your sending circuit after you change your oil filter for added insurance in case it's clogged or defective. But go for the 100% approach. Don't just change the filter and cross your fingers. We're talking oil pressure here.