[2020] Road snapping not working as it should...

You can't do what you're trying to do without mods. It would create 2 intersections really close to each other, which is a bad idea in this game.

You could use the curve tool to have it curve just at the end and connect to the existing intersection. Or you could just leave it as a dead end like it already is. Those would be my recommendations here, as the 2 super close intersections would cause you traffic problems.