I've noticed a stigma lately (yes, another one), that people refuse to go out alone because they feel as if they wont have a good time. While I agree that it may be safer to be in groups in certain situations, being alone is not the worst thing! I mean really, don't you get tired of people?
Let me start by saying, confidence is key. If you look worried or lonely, people take note. It's not the happy person that usually draws your attention, but the one that looks like they feel out of place. Otherwise, you look like just another person having a good time at dinner.
Stop looking around like you're uncomfortable. While you may believe it makes people think that you're waiting on someone, you're just staring at people, and that makes them stare at you. Don't be the weird guy that watches people eat all night.
Learn to enjoy your "me-time." Take advantage of the time you have to yourself. It's one of the simple pleasures in life that you wont always have, as parents can attest to. Let it be a time when you organize your day or let loose and relax without having to worry about others' expectations of you.
Once that you realize that you don't need a second shadow everywhere you walk, you have the freedom to let the FOMO go.