Using 3.5 mm cable to connect load cells?
It depends on the gauge of wire used within your cable.  In order to support a 350 Ohm load at 3.3V, your wiring will need to supply a current of around 10mA.  The signal wires from your load cell will see virtually no current, as the input impedance of your ADC should be very high.  You'd have to try pretty hard to find wiring thin enough that it couldn't support 10mA.

A TRRS cable isn't a standard audio cable, though.  Unbalanced stereo cables are typically TRS, as are balanced mono cables.  A TRRS cable is probably some sort of combined audio/video cable.  I've seen them used with mobile phones to output stereo audio with composite video.  

Your oddball audio-video cable will probably work just fine, but I generally caution people against using standardized plugs for applications other than their intended use.  For example, you shouldn't use a 3-prong edison plug as a stereo connector for your Hi-Fi receiver's auxiliary input - someday, somebody is going to come along and plug it into the wall outlet and blow up your receiver.  If you're already going to splice your own connectors on a cable, I would pick a connector type that clearly has no other purpose than that for which you're using it.

Also consider that a female 3.5mm TRRS connector looks outwardly identical to a female 3.5mm TRS connector, but if you plug a male TRS connector into it, you will short a ring and a sleeve to each other.  The same is true with a male TS connector.  Even if you wire up the tip as your 3.3V supply, you could still run the risk of raking your 3.3V supply across the rings and sleeve of the connector as it disconnects, which will happen if anyone trips over the cable while everything is powered up.  

I'd use a locking connector of some kind, preferably something that doesn't short pins to each other while being connected or disconnected.  Depending on how durable and/or permanent you want this setup to be, you could go the cheap plastic route and get some cheap Molex connectors, or go the expensive route and get some metallic circular DIN connectors.