Is it possible to change the voice alexa is using within my skill?
i.e. the user asks
Alexa Ask Car Washing when the next available appointment is?
and have Alexa respond with a voice that matches my carwashing brand?
Yes, it is possible.
You can use SSML Tags in output speech response of your skill to achieve this.
You can
For Example
<speak>
Here's a surprise you did not expect.
<voice name="Kendra"><lang xml:lang="en-US">I want to tell you a secret.</lang></voice>
<voice name="Brian"><lang xml:lang="en-GB">Your secret is safe with me!</lang></voice>
<voice name="Kendra"><lang xml:lang="en-US">I am not a real human.</lang></voice>.
Can you believe it?
</speak>
Learn more about Using SSML Tags HERE