Bikes were not dangerous before disc brakes. They don’t all need “upgraded”.
There’s lots you can do to improve your braking.
New cables and housing, lube all the pivot points. Clean the rims with 600-1000 grit sandpaper and alcohol. New pads, Cool Stop Salmon are loud and work great. Probably and afternoon of fiddling and $40. Less than $100 for a shop.
This would be your only real option and good news, it’s a good option.
To convert to discs, you’d need to stretch your frame to 130mm and find the mythical 130mm QR disc hub. Then get a mount welded or brazed to the frame.
Up front is a little easier, but a threaded 1” fork with disc mounts is not going to be easy to find. The 100mm QR disc hub does exist though.
TRP makes some great road lever/calipers. That’s another $300.
I think the conversion would be close to a grand and might not work very well.