What is the process and processing time for applying for the LTR visa from Washington DC?

Where you apply from doesn't mean anything. It's an online application processed by the BOI. You don't mail in anything.

Once you are approved you can decide if you want to pickup the visa in Bangkok at the Chamchuri Square immigration office or if you want to apply from your Thai embassy via evisa. The e-visa will be significantly more expensive because the embassies get to set the price rather than the 50,000 baht if you purchase directly from immigration at Chamchuri Square.