Interesting question...
I have rarely hesitated to go to the course alone, whether hoping to find a game or intending to fly solo. I have met many interesting people over the years, and find both the social aspect and friendly competition rewarding. Having said that, I have had some of my most enjoyable rounds on my own, both while traveling and at home. This past Summer, I regularly went out Sunday evenings for a solitary game, and loved every minute. I suppose it is the closest thing to meditation for me, and I love being able to walk the course at my own pace, usually finishing in under 3 hours.
In terms of dining alone, I must admit to being pretty self conscious about it. I am not exactly sure why, but I typically go the bar route (or even just order room service when traveling) when eating out on my own. The only thing I might enjoy as much as playing a great golf course is enjoying a good steak, so perhaps I should get over it.