Hey @Huz88,
I have found that forced conversation is less genuine and even more awkward that silences. Have you tried overcoming this by chatting up random strangers at a bar?
What if you tried treating your approach as a social experiment instead of overcoming your anxiety? What I mean by this, is there is no wrong in what I'm about to describe (you can't fail or say the "wrong thing"). Go into a restaurant that has a bar, not like a pool hall but a place where it's like a nice vibe to the place -- sushi restaurant at happy hour, sit one chair away from someone, and start a conversation like "What is good here? I've been here 100 times (or this is my first time here) and I'm in the mood for something different. What did you order?" And after that, see where the other person carries the conversation. If they continue to ask you further questions like "Are you from around here?", then proceed to carry on the conversation. If they drop the conversation, let them drop it. Let the burden of carrying the conversation be on them during your experiment.
There should be no burden for you to carry a conversation.