If you approach it as no big deal, others usually do so too. That's my experience.
You know, on a side note, I get migrane headaches that don't really give me a bad headache, just a little headache, but instead it effects my vision or speech (depending on what it chooses to do at the time, doc says they're harmless and will go away like any other headache).
So, I'm in the McDonalds drive thru and I ordered one basketball net and a golf cart. I could NOT say "Grilled Honey Mustard Wrap and a coke", it just kept coming out as I want a basketball net and a golf cart. It was hysterical. I could have choose to be frustrated as heck, I could have went home and cried, I could have asked "why does this happen to me", or a hundred other things. Instead I pulled away, write down what I wanted and when it was my turn at the speaker again I drove all the way to the window and handed the lady my order I wrote down. The next day I went back to the drive thru to order again. When I was paying I told her I get headaches that effect my speech at inopportune times. She smiled, said it was OK, and took my money and told me to proceed to the next window. No biggie.
When it effects my vision I go blind on 1/2 of my perifreal vision or more. When that happens I have to simply must lay down till the headache goes away (or vision comes back actually). When it happens when I'm driving I umm, change course immediately and find a place to relax in a parking lot or go home to lay down more comfortably (instead of reclining the seat) if I'm close enough. It takes about 1/2 hour of rest to make it so I can see again.
So, you can make a choice. When things happen you can be frustrated and embarassed, or you can do what I do when I stutter and make it THEIR problem to try to understand you and think of it NOT as a problem at all, but instead something that simply happens.
Anyone that gives you a hard time, or adverse time, about your stuttering when you do so should be embarassed of themselves for not having the social skills themselves to be considerate to someone with a "handicap". I know you are not handicapped, but when you stutter (like me) you kinda simply are.
They don't know you're nervous, you could legitmately (which you do) have a stuttering problem. Anyone that doesn't accomodate makes it their fault they are an a$$shole and not yours for stuttering what so ever. What if you were a war hero that saved 100 other troops by taking a bullet to the head and survived? What if the result of that effected your speech with a stutter. What if you pushed an old lady out of the way of a speeding semi-truck that lost it's brakes and you hit your head, survived, but stuttered as a result? Would you feel differently? You shouldn't, because any reason you stutter is reason enough for you to be treated with respect.
Remember that.