The single click you now hear is the solenoid pulling-in, but there's not enough battery to turn the engine via the starter motor. That is a classic low-battery scenario. The way the starter's performance slowly worsened makes me question the battery or the charging system.
Have you measured the battery to be certain it is fully charged? A new battery cannot remain fully charged if the charging system (alternator) isn't working properly. A healthy battery will measure 12.6 to 13.2 volts across the terminals. When the car is running you should measure 13.6 volts (mechanical voltage regulator) to 14 volts (solid state regulator) across the battery terminals.
-G
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