I need some help, or more so, advice on what I should do with my bass.
A while back, I bought a set of strings (Ernie Balls) and slapped 'em on my bass
only to get the E string resting on my frets and basically, all you could was
a muted note until you pressed down onto the 5th fret.

I put back on my old strings and it was good to go, no worries.
That was about 5 months ago, probably less, but anyways.. I was playing
a show last night and we tuned down to Drop C and I could already hear it muting
like before. When I'd hold my bass to where the body was parallel or slanted to the
ground it would mute, so I would need to hold up (standing position, proper holding tech., whatever). Which was fine, not a big deal, I figured it was because of me being
in Drop C.

Then however, I came home and was going to try learning an All Shall Perish song
meaning I would need to go to Drop B, so I did. However, when I did I just got mute
notes all the way, even when I tuned back up to E standard it was just straight mute
notes on almost all strings, so I just took them off.

Any advice on what to do, or what the problem could possibly be??

I had a friend suggest the nut was worn out (the strip/block resting above the neck on the headstock).

Could I raise my strings up?
If so, how would one go about doing that?

Any help would be appreciated, so thanks in advance!!
raise your action.

Go to your bridge, find the allen wrench holes near the strings, and turn them so the string raises itself up, then tune the bass back. You'll be sharp, just as a heads up. Do 1 string at a time , don't just raise them all, cause some of them probably don't need it.

Could be your truss rod as well. Put your head down by the bridge and look up the neck...is it bowed? Which way? It should be bowed foreword a bit, like the strings are pulling it down. (well actually, they are..)

What I'm thinking is that the new strings, as well as tuning drop C, allowed it to bow back, and it settled like that. If so, you'll need to adjust the neck relief(truss rod). It's not difficult.
Nope, no sig here.
Yup. Try raising your action first though, that'll be faster, and let you know if it is your truss rod. You shouldn't need to raise your action more than about 1-2 full turns, and if you need to do more than that then your truss rod needs some tweaking.

If that's the case give your strings some slack, take off the truss rod cover, fit an allen wrench in the hole for it, and give it about a quarter turn to the right I think, if you're looking down the neck from the headstock. That should be enough, then let your wood settle for a day or so to let the wood sorta reset itself, and get used to the new tension or w/e.