I do not think is safe to connect to a regular outlet without knowing the load on that cable run, is better if you have a dedicated cable run for those appliances.
Your statement needs to be clarified, do you need a 120v washer/dryer wall unit that connects to a regular wall outlet?
Great
Best thing to do is get a good electrician to change the outlet to code & then get the appliances from Lowes.
Good luck. I doubt you will find it. No dryers these days has a regular plug. All have the funky special plugs.
Take a look online also look at BestBuy and Home Depot
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