Hmm, an interesting question to be sure...
I didn't realize it until recently, but once upon a time in my life, it was probably very obvious I was needed friends and while I didn't beg per se', I think it's possible I came across as a bit desperate. I don't know that for sure as I've never asked anyone that; but looking back at my own behavior, I have to wince a little because it is possible that the reason I didn't make many was because people picked up on my desperation.
There have also been times that I've felt like I was practically begging for the attention of my existing friends. That has made me feel useless and worthless on occasion. Then I start the downward spiral of thinking that maybe I am worthless and how could those friends possibly like such a pathetic worthless person like me and well, on and on.
Over time though, I've mostly broken myself of that thinking. It helps that I have more friends in my life now so if one is unavailable for whatever reason, another probably will be. But there was a time I only had one true friend that was mostly available and I probably drove him about crazy at times. But he is the soul of patience and was there for me through that low period of my life. Now that I have more friends, I don't feel so starved for attention because now I am getting all the attention I need from multiple people and that has gone a long way to help my self-esteem.
So now, on those occasions that all my friends happen to be busy (right now as I type this is one of those times), I am okay with that. I am much more confident in how my friends feel about me and know that even if they don't feel like chatting or are busy right now, they will be soon enough. I just enjoy the alone time as I have it now as it helps calm my spirit.
Just try not to overthink making friends. Just let them happen naturally. If you come off desperate, you're going to attract those who are manipulative or will think you're gullible (even if you're not) because they'll see you as an easy target. Just try to relax and smile. I know it's easier said than done, but it will help you feel better, even at those times that it feels like life sucks.
I hope this helps.