Jim,
mrx is right - those pipes are not in good condition and any changes are likely to set off a chain of nasty problems.
Here, two possible solutions which occur to me:
1) Those Y-connectors which people use to hook up a portable washer to a sink can be bought at any Home Depot. They aren't expensive and work just as well in the other direction.
You'd only have to stick around by the Maytag long enough for it to fill for the wash. Rinses and spray rinses are only cold water in Maytags.
2) If you really want to risk it, those union joints are the place to make the change. Holding onto the top of the union fitting very tightly so that the galvanized pipe won't twist at all, loosen the fittings one at a time and remove the faucet.
Get two new faucets, if you can get ones that will fit into these union joints, great. If not, pre-fit the new halves to the faucets first. When you mount the faucets be absolutely certain that the pipes are neither twisted nor put under any load whatsoever.
Secure the new faucets to the wall, not the pipe behind them.
I'd go with my first solution, frankly. Oh, and if you do change things out, don't use the 0.66 roll of Teflon tape from the bins at Home Depot, either use proper pipe dope or the higher quality Teflon with NSF certification.
Good luck!