Great to hear you found a properly wired, grounded circuit!
You can not use the L1 EVSE that came with the car on a 240V circuit. You will need to purchase a 240V EVSE (Level 2 or L2) to charge your car at 240V. The included L1 EVSE charges at 1.2 kW but you could charge as quickly as 7 kW (240V, 30 amps) with an L2 EVSE, if your dryer circuit is 40 amps. You NEVER want to pull more than 80% of the circuit rated amperage. If you have a 30 amp dryer circuit, then the maximum continuous amps you can pull is 24 amps, but that still works out to 5.76 kW, which is almost 5 times faster than the L1 EVSE.
Do NOT get an adapter plug for your L1 EVSE as 1 as it is not designed to take 240V, as far as I am aware, and you don't want to start an electrical fire.
Yes, you can use your dryer plug. There are products on the market now that allow you to plug both your dryer and a L2 EVSE into the dryer plug but the device will give priority to the dryer when drying clothes and otherwise let your car charge when the dryer is not running. These devices cost around $400-$500 and is a great idea for you to avoid using an electrician and also not taking up more space in your breaker box. You do not want to be constantly plugging and unplugging devices on a 240V receptacle because it will quickly wear out and lead to sparking and potentially a fire. It is much safer to buy a smart sharing device (below) so you can "set it and forget it" and not need to repeatedly plug and unplug devices.
Tom Moloughney reviewed these devices (on you tube) for using an L2 EVSE with your dryer:
Look at these: