Having travelled for business early in my career, eating was always somewhat of a challenge.
Tried staying at hotels with kitchens, but was such a bad cook that I fell asleep with a pan on the stove, it melted, and set off the hotel fire alarm. It was super embarrassing getting woken by security and having everyone at the hotel glaring at me for waking them up. So much for staying in to avoid looking alone!
So it was full steam ahead going to eateries alone, and making it work for me.
At first you are self concious, after awhile it turns into a challenge to counteract your irrational fears. Know that those around you all have their own problems, and may be with people they really don't want to be with. You on the other hand, are unencumbered by unwanted guests, and can use your god given charm to find a wanted guest.
Yes, believe it or not, being alone is a huge advantage when meeting others and opens doors to you that would have never existed otherwise. With just a little tact, and clean clothes, and a smile - nothing should stop you from accomplishing your goal (whatever your goal is!).
If you never ask, you can expect to never get. Its truly amazing what you can find by asking.
Just my experience, from a really shy person who has to work at this stuff.
michael