If you have enough space in your existing outlet boxes (including box fill calculations), you can cap the K&T wires and simply tuck them out of the way. I'd also recommend electrical tape to ensure that nothing accidentally moves or gets exposed, especially since this is a permanent disconnect.
If there isn't enough space, you can use a surface-mount junction box over the existing outlet box, or, as I've recently done, a raceway outlet box. You can cap and tape the K&T in the old, recessed outlet box and pull the new cables further forward into the surface mounted box. When you're all done, cleanup is as simple as removing the surface mounted box and tucking everything back into the recessed box.
6/3 NM-B is not wire, it is cable. Cables contain multiple wires. NM is fine for running through ceiling joists in dry areas where the cable will be protected. If water is involved in any way, you need UF cable.
Cable can be run through conduit, but only if you are a masochist. It will be a rather difficult pull. You will need much larger conduit, because the conduit fill is calculated by looking at the widest dimension of the flat cable, and imagining a round wire that diameter.
There's an exception that allows this imaginary wire to fill most of the conduit if it is the only cable in the conduit. But that only makes it an even more difficult pull, and paints you into a corner as far as any future use of that conduit.
You cannot unsheath cable and use the individual wires. That is because the individual wires do not have individual wire markings to disclose their type and grade of insulation, and generally their insulation is not tough enough to survive any type of use outside a junction box. They are allowed as short pigtails entirely inside junction boxes.
If you find cable whose interior wires are marked as some sort of wire that is legal for use in that application, then have a field day. This is rare but not unheard of, talk to your local electrical supply.
You cannot do a continuous run with new wire because you cannot use loose wires in through-joist wiring. You need to either fit conduit in the joists, or use some sort of cable rated for such use.
If you run all metal conduit, that is one less wire you need to carry, as the metal conduit serves as the ground wire.
Before you decide on the romance of continuous wire runs, count your bends. More than 360 degrees of bend between access points is illegal. More than one makes pulling exponentially harder.
It is very easy to build a conduit route that is so difficult to pull that you end up needing to call a professional to do it. Professionals don't like doing a small part of the work, and will want a big chunk of the total job. Access points are the key to avoiding this, but of course, access points are the enemy of "a finished look" for some reason. I don't know why, the Victorians had no trouble fitting adorable cabinets where utility access was needed. It is a 20 minute job to change my tub faucet. Really.
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Wires exiting have to be counted too. You'd need to add 4 for the two NM cables. Conduit lock nuts, NB snap and outside clamps are not counted in the fill calculations. The grounds leaving with the NM cables don't count either because you've already counted a ground.