The waiting list option mentioned earlier in this thread does work and is available to foreigners.
However, it is only available on the Chinese language version of 12306 (web and app), not the English versions or on trip.com.
It looks like China Rail is heavily using the waiting list to manage tickets, so it's an option to consider when you see "Sold Out" on the English sites.
I used it earlier this week.
I booked a confirmed train that I could take, paid using an int'l credit card on 12306.
Then I waitlisted for a 3 trains I would rather take, also paid for the maximum fare that could be charged for any of these tickets using an int'l credit card.
The waitlist cleared relatively quickly - I received a refund of the difference between first class and second class (a second class seat confirmed).
I refunded my original ticket. As it was still >1 week out, I got 100% of the amount back.
They even give you what % success rate to expect.
The only thing I have NOT been able to do is buy a ticket for myself and a Chinese national on 12306.