Do you do stuff solo at home? Movies, eating out with a waiter not counter etc, not just running errands.
If it's not something you are comfortable doing on your own at home I'd start there before traveling. There's nothing weird about doing anything alone. Sure sometimes, though not as much in the past few years as over a decade ago, telling a hostess one would sometimes get a reaction. I think it's more common now but also I don't care, I just want to eat.
I'm also an only child so I often played by myself and just learned early that if I want to do something and I don't want to miss out on it then I need to go alone or be ok with not going. Am I still annoyed with myself for not going to see Romy & Michelle the musical when it came to Seattle, yes but at the time I just didn't want to deal with going by myself. But I also skip stuff in general when it's in Seattle because it's become so annoying/$$ to park in and the light rail takes too long.
The other thing to remember and it holds for at home too, when you are out, you'll probably never see these people again so it doesn't matter what they think and no one will probably say anything anyway.