i don't have recommendations, i can just list out stuff i learned since i moved recently
there are two types of movers local vs. national. local guys will be good for a 1hr to 8hr move, while national guys are good for "long" moves.
sc to las vegas i'm sure is considered a long move (31hours by google maps) so you're probably looking for national.
they'll charge by 3 ways
charging by time of course depends on the total move usually pick up/travel (to your destination and from is counted) and delivery. this is probably not for you
volume (i still don't understand how they measure this) the amount of "stuff" you have
weight usually you're stuff will get weighed at every weight station.
your situation seems to be good for "weight"
moving companies are beyond booked in the summertime and rates are almost tripled, that is when most everyone moves and i've been told ppl even book 6months in advance just because of it. i moved in the winter (the deadest time) and everyone wanted my business.
if you need boxes, make weekly trips to your local walmart/target/supermarket and talk to the manager and ask for boxes. there will usually be stacks of boxes (cut so that they can be folded) just sitting out there.
otherwise you can buy boxes, uhaul.com is a good place, buy over 50bucks and they ship for free, also you can "over"buy from uhaul and any extra boxes you have left over (that aren't used/touched) they'll buy back from you.
if you need bubblewrap (you will for mirrors/tvs) goto furniture stores, they'll usually have tons of it waiting near their trash
or you can buy 100ft for like 20bucks
Amazon.com: Duck Brand Bubble Wrap, Large Extra Cushioning Bubbles, 12 Inches x 100 Feet (1061909): Office Products
i like duck brand stuff
just stuff i learned from my recent move, hope it helps.