wont start in cold weather, clicking/buzzing sound from under dashboard, new battery/ good altenator
Check if the full voltage is getting to the solenoid when you turn the key to the start position. If no voltage makes it to the solenoid, try to isolate the problem beginning with the key switch. If partial voltage appears at the solenoid, look for high resistance connections or high internal resistance in the battery. Problems with the solenoid and starter themselves can be confirmed or ruled out by applying battery voltage directly to the solenoid with a remote starter switch (purchase, borrow, or make your own).
This should go without saying but be careful to avoid short circuiting anything when working with jumpers (remove metal jewelry), wear safety glasses, and have it in park with parking brake set and your body and limbs in a safe place when applying power to the starter.
On a car this old, I suspect the starter to be the problem.Remove the starter(Denso) and replace the 2 copper contacts and plunger inside the solonoid housing(solonoid contacts are available on ebay for $10).Bench test it to see if the problem is resolved.
Does the engine turn over (rrr-rrr-rrr) when you turn the key to Start? The click sound, if in the dash, could be a fuel pump relay. The engine should still turn over, but not fire up and run. The click could also be from the solenoid on the starter, under the hood. That sound usually means poor battery or connections, or a malfunction within the solenoid or the starter. How much work do you want to do on your own?