Along with cleaning the connectors, try this.
With the watch not connected, hold three buttons down (light, down and start buttons) for 15-20 seconds and try charging again.
I just finished chatting with Erica from Garmin Support Centre. The chat took less than 10 minutes and resulted in them sending me a refurbished watch, and I'll send in the defective one when I receive…
Mine wouldn't charge at ALL. I did your trick thinking it wouldn't work as there was no power left but it worked. Thank you!