Ah, a fellow support! I was in a very similar situation, except level 30 and playing ranked.... Good on you for trying to solve your problem earlier on. XD
Right now, I'm still better at support than any other role, but I've started to branch out. Maybe I can give you suggestions of what worked for me.
As xCrosshairsx said above, practicing farming in a custom game is extremely helpful to improving your farm. Many guides suggest starting out by yourself, then when you're comfortable farming like that, adding a bot to simulate an enemy's harass. You can also 1v1 against a buddy. A common goal is to get around 7 cs/minute. Keep in mind this is hard, and I usually am only able to hit 4-6 cs/minute in my normal games in my most comfortable lane role. For perspective, most pros hit around 10 cs/minute.
I would suggest trying to learn ADC next. As a support main, you're already very familiar with bot lane mechanics, so that would give you an instant edge in your games. Also, (in my opinion) last-hitting is easier with an autoattack than an ability cast, which you would most likely be using to farm in mid. Caitlyn and Sivir are good ADCs to try out.
As you learn new roles, keep in mind you will be learning new champs as well. This will make everything a bit harder, but don't let it discourage you. As you enter a game, make sure you are familiar with your champ's playstyle and kit. For example, Caitlyn and Sivir both tend to outtrade their opponents in the lane phase and have good waveclear.
As for being "forced" into a role, I can't recommend team builder highly enough. The waits can be long and the comps are sometimes silly but it's the best way to ensure you're playing the right champ in the right place. It's a lot less stressful than blind or draft pick, and, it's likely that some of the other players in there are trying something different as well or just trying to have a good time.
Diving? Roaming? Recalling? To be perfectly honest, given that you are at a level under 20, I would discourage diving and roaming in most cases. It's not because you aren't skilled enough, but because you don't currently have access to the runes/masteries that are so useful in diving and roaming. In your case, these two things are much more risky. That being said, diving and roaming are two skills that you will definitely need to develop as you continue playing LoL. I would suggest familiarizing yourself with champions that are good at these two activities - for example, Nidalee likes to roam because of her ability to pounce long distances and her brush passive.
As for recalling, there are a few "rules." In a solo lane, you should try to push your lane first to prevent the enemy from damaging your turret when you're gone. You should also try to quickly farm then recall when your opponent recalls, because he will soon be back with better items. Duo laners should try to coordinate their recalls to avoid a 1v2.
I also would recommend, if you don't already, watching the LCS and pro streams on Twitch or Azubu. This is a really good way to learn about strategy, mechanics, tower diving, champions, shotcalling, anything. Professionals are a much better source of advice than me haha!
And, above all, do keep your chin up. It's not that you "aren't good," you just are unfamiliar with most roles. And you can change that by practicing! If you mess up, don't think negative thoughts about yourself, but remind yourself that mistakes happen and try to learn from it so you can do better the next time. Always reply to friendly teammates with a "gj" or "ty" and always reply to unfriendly ones with the good ol' mute button. ;)
Well, hopefully this wasn't too long of a reply. I hope what I said helps you, and I wish you the best of luck, supporting or anywhere else!