I still have many issues carrying partly due to my weak mechanics
Well that has many implications.
Carrying is not about out mechanic-ing your opponent(s) and getting all the kills, it's about learning how to turn a snowball into objectives into a bigger snowball and so on.
Think about it like this: It is very hard to carry with Nunu unless you're a very smart and crafty AP Nunu main if we're talking strictly about getting kills. However, Nunu has a few rather significant tools if you think about him from an objective standpoint. His Q is one of the most fantastic single target skills in the Jungle, making him one of the best champions in regards to dealing with Epic Monsters. His W grants pretty solid %s for ADCs to utilize, which helps them clear objectives for you or keeps them alive longer. His Ultimate is a titanic zoning tool and a pretty universal red light for the enemy team, meaning that if he stands between an objective and the enemy team, he can effectively shut them out of something extremely important. You can carry hard as hell with a Nunu without ever getting any kills simply by rushing CDR to abuse the powerful base effects of his abilities, and enough tankiness so that he can comfortably zone the enemy team without putting himself at risk.
Every champion has some tool that grants them a win condition. Nunu, as I described, has 3 win conditions of Epic Monster Control, Supporting the Damage Carry, and zoning enemy teams. If you can learn to recognize these win conditions and play to your champions strengths within those win conditions, you will learn how to carry hard.