The normal domestic supply in the USA is 110 to 120 volts and uses a socket for a plug having two vertical flat pins. Using an adaptor, your hairdryer will work with this mains supply if it's been designed to do so. However, UK TV sets, vacuum cleaners, and washing machines, etc, are built for use on 220 to 240 volts, and won't work on 110 to 120 volts.
However. More modern dwellings in the US may also have a 220 volt outlet. The plug for this again uses two flat pins, but one is vertical, and the other is horizontal.
With appliances using a lot of power, like a washing machine, tumble dryer, or room heater, it's unwise to use a travel adaptor - they're not up to the job and can easily overheat. Instead, simply have someone change the plug for a US style one. If you are going to live in the US, there's no point in keeping UK plugs on electrical items. Have them all changed when you get there.