From your post, I think you already know Why you feel uncomfortable eatting alone. You said:

It sounds like I dwell on what people think of me, right? If so, how can I overcome this?

So how do you stop dwelling on what others think of you? Focus on something else, like how good the food is. Or how brave you are to eat alone. It isn't weird to feel that way, I get it. I used to think it was wiered, until I did it a few times.

I have learned to not worry (anxiety) about what others think. I've had to get over it if I was going to get on with my life. It is all about what you think about yourself. But you are worth it, you can do it, rethink the situation.

I enjoy going to eat alone now. I get to do 'People watching' and don't worry if others are watching me. They are probably wishing they could have some peace and quiet away from the hub bub of everyday life-having to have a meal with the person they are with!!!!! When I go shopping, also alone a lot of the time, I take a rest in the food court. I sit where I can watch the other mall walkers, shoppers, teen agers acting silly, mall workers rushing to get a quick lunch, the divas, the free fashion show of those who really dress up to go to the mall....so many different people.

One thing you might try the next time you choose to go eat alone is to try to remember a person who catches your attention and image that person as a character in a movie or a book. While you eat, write a good description about that person or group of people, and what you think they do in life, if they have a dog or cat, where they live based on how they look.

Use distractions like that rather than feeling self-consious about being alone. You are worthy of going to a restaurant and feeling comfortable in your own skin. You just don't believe it. As for the 'negative thinking' you have created in your head about the waitress and the tip thing. If they are good at their jobs, give them a nice tip. If they really are rude to you, it's them, not you. They don't deserve a tip. Changing your thinking is something that you can do. You can't make them less rude.