I know exactly what you're facing here. I've finally worked myself up to finishing my placements (much to my enjoyment I made it to silver 2) and have only played one game since.
The first thing I would recommend is getting a group of people to play with. I absolutely despise playing with PUGs, but I have a group I am very comfortable playing with, even on my off days. I've played with them enough that I am actually quite confident in my ability to play. You should try this, find a few friends to practice with and keep practicing until you know what your strengths and weaknesses are.
Once you're confident in your ability to play things get a little trickier. You know you CAN do it, but now you have to convince yourself TO do it. I've gotten into the habit of playing a game of ARAM, Team Builder, or a PBE match since all of these are typically less serious queues. A warmup match is always a good reminder that you CAN do it, particularly if it's in a relaxed environment with strangers. It doesn't always work, but it's a start.
I usually listen to music that will get me in the mood for a challenge while I play. You'll have to find the music that works for you, but I usually use soundtracks, or even the music from the team builder queue. Music can alter your emotional state far easier than encouraging thoughts can.
The only other thing I can think of is if you have a bad game take an immediate break afterward, but force yourself to come back. You don't have to play a game right afterward, or even for the rest of the day, but making a habit of playing a normal the day after a bad encounter will help reduce that anxious feeling after a while.
Just keep in mind these tips are not an instant fix, these things take time to overcome. I'm still dealing with this crap myself, but with a bit of persistence I've made some progress. Just keep at it and you will too.
-1 piece of pie from SOLO Q
