The O2 sensor tells the computer how rich/lean the mixture is supposed to be. At cold start, extra fuel is introduced which may cause fouled plugs to become wet and not fire.

Replace the O2 sensor as soon as possible. You want to wear gloves anyway...

If the heater core is not leaking very much, wait until the end of this month but before your trip. Give yourself 2 hours for the job.

Check your vacuum lines, all of them. If you have a turbo, verify that the 3 lines to the turbo are not cracked. A crack in any line will also cause the engine to run rich.
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Keeping it running is better than buying new