The immediate problem is the turn signal "cam", which is the topmost part of your turn signal switch. It's a sort of ear-shaped piece of plastic that rotates slightly clockwise (for a right turn) or counterclockwise (for left) when you move the turn signal lever. This moves hook-shaped pieces of plastic at the upper (right turn) or lower (left) end of the cam into striking distance of a metal finger clamped around the steering wheel's underside shank. When the steering wheel rotates so this finger contacts the cam ear, it pushes the cam (and therefore the switch and lever) back into the neutral/no-turn position. The plastic gets brittle over the years and breaks off, then the switch doesn't cancel any more.
You can buy a replacement cam for twenty-odd dollars, such as a NAPA Echlin DL-6123. It will fail quickly, because they are made of a cheap grade of plastic. A long time ago (most of 20 years) you could buy a cam made of much better material from David Koldos. Perhaps you still can. If you can, and you do, your immediate problem will be fixed.
However, your turn signal switch is probably half a century old. The cam isn't the only thing that wears and fails. The detents in the switch body get loose and sloppy so the lever no longer snaps into/stays in the left/neutral/right positions. The contacts and shuttles inside the switch wear down so they no longer reliably direct current to the various lights; you start to get symptoms like front signal works but not rear (or vice versa), or brake lights sometimes don't light (their power goes through the turn signal switch). Before that happens, the signals and brake lights grow gradually dimmer because of increasing resistance in the turn signal switch.
You might want to fix and prevent all the problems all at once with one of these, instead.