I can't afford 10-3 right now but I do have two runs of 10-2 the washer/dryer says 120/240 on the back plate,is old and came with a 3 prong cord attached. I know this is not up to code but is it possible to use 1 of the cables for the power and ground and the other just for a neutral?
You can't put hots and neutral in different cables - there are very good technical reasons why with AC power everything going to one circuit has to be in a single cable. That rules out using three insulated and one uninsulated wire to handle everything - that gives you the right number and type of wires but not all in one cable, so that just doesn't work. See NEC 300.3(B). There are exceptions, but they don't apply here.
The old style dryer setup was 3 wires, hot, hot and neutral. With an allowance to connect the dryer ground to neutral, so that you could make use of an existing ungrounded cable. That was outlawed for new connections all the way back in 1996.
Conceptually there should be little difference between "ground from the dryer connected to neutral" and "neutral from the dryer connected to ground", but there are a bunch of problems with this. Aside from being a blatant code violation, this makes a bare wire neutral, which is never supposed to happen on a branch circuit.
Plus people can die from 3-wire dryer connections (even without a misuse of 10/2 as you suggest).
If you have an existing 3-wire (but no ground available) connection and an existing 3-wire receptacle, then it is code legal (but not advisable) to plug a 3-wire dryer cord into the receptacle. But every dryer made since 1996, and many made years before that, has an easy way to switch to a 4-wire cord with neutral and ground totally separate. And since you don't already have a 3-wire receptacle wired up (if you did, you wouldn't be asking the question about 10/2), you are not allowed to install one.