Greetings from NoVa sir - as long as South Carolina has reciprocity with Virginia - then you should be fine ... but only in Virginia. I'm not 100% certain, but I'm 99% sure you cannot take a handgun into DC itself. That's a little tricky since Reagan Airport is considered part of DC - hopefully you are flying into Dulles. Virginia CC rules are typical - no public/gov't buildings, schools or churches. I don't drink so I'm not sure about alcohol - so I would refrain from carrying into a lounge or bar. Since you are out of state, if you are stopped, I would notify up front. I assume you've already been to HandgunLaw.us - but here is the PDF just in case:

http://handgunlaw.us/states/virginia.pdf

HandgunLaw.us says Virginia honors all other states - but it's just a website - you may want to do some additional research as the other poster commented that VA does not honor other states. His statement was true for several months last year - but I believe that has changed. Probably my most important comment: do not even think about driving one inch into Maryland with a handgun. They scan plates for CC holders from other states and will stop you - one confirmed case last year for a person from Florida.