When you mean "starting it once a week, do you mean simply starting it and letting it run, in place, for a few minutes, then shutting it down?
To me, it sounds like you have 2 or possibly 3 problems. First, you have an intermittent problem with the starter engaging. Since you just replaced the starter, one can assume from your symptoms that you either have very poor battery connections or a defective starter solenoid.
The next suspect, especially since it's Seattle monsoon season is poor ignition wires and possibly distributor cap. If you open the hood at night and have someone turn the engine over and it looks like a lightning show that's a good sign that you should replace the ignition wires, clean the top of the coil and distributor cap, and check your point gap/dwell to insure you have a good crisp spark.
Next on my list, once the other two have been addressed and should they not fix the problem, are some fouled plugs from not giving the engine a chance to thoroughly warm up and burn off the deposits formed while running on part choke. If you feel like spending a couple bucks and replacing the plugs, fine, but if you can get it started and run it around for a half-hour they may clear up without a problem.
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