I will be flying out of JFK T4 on SQ to SIN (via FRA) and am trying to figure out the best way to get to JFK from Washington DC.

Which option would be best for me?
1) Fly Delta from DCA to JFK. I would have to pick up and re-check in my luggage.
2) Take a long ride to IAD and fly UA to JFK. Given that UA and SQ are both on Star Alliance, am I able to check in my luggage all the way from IAD to SIN?

I guess there are no direct trains to JFK?

Appreciate your help!