You can put the name before the job, or the job before the name. In either case you would be better with a comma:
"I don't want to embarrass myself in front of the stylist, Sally."
"I don't want to embarrass myself in front of Sally, the stylist."
(Actually, in the second case you could miss out the comma, but that would change the tone - you would seem to be calling her "Sally the stylist", rather than calling her Sally and noting that she's a stylist)
The difference between the two is subtle. In the first, you don't want to embarrass yourself in front of the stylist, whose name is Sally. In the second, you don't want to embarrass yourself in front of Sally, who is a stylist. You would be more likely to use the second where you know the person socially, rather than just professionally. However, it weakens to link between the whole sentence and the fact that she is a stylist. You can, however, use an adverbial of reason to indicate the link between your "don't want" and the fact she's a stylist:
"I don't want to embarrass myself in front of Sally, because she's a stylist."
That isn't really what you mean, though. It's not her being a stylist that stops you wanting to embarrass yourself. You think that you're more likely to be embarrassed by your clothes because she's a stylist. So we make it a bit more complex - we need a whole, standalone clause, but to associate it closely with this sentence. We want a semicolon.
"I don't want to embarrass myself in front of Sally; she's a stylist, so she might think my clothes are poorly chosen."
"I don't want to embarrass myself in front of Sally; she's a stylist, so I'd better pick my clothes carefully."
Of course, that level of detail might be implicit, so we can actually reduce that second clause right down to the essential factor and hope the reader/listener gets what we're saying. This foregrounds the fact she's a stylist and relates it to the desire to avoid embarrassment. However, it does still leave it ambiguous as to whether you're noting that you need to do different things to avoid embarrassing yourself in front of Sally (i.e. pick clothes carefully) or that you don't like the idea of being embarrassed in front of a stylist. Whether this is ambiguous, or whether ambiguity is a bad thing, depends on context, and your intent.
"I don't want to embarrass myself in front of Sally; she's a stylist."
I hope that helps.