I'm using an HDMI-to-DVI adapter to connect it to the monitor where I've gotten it to work, but that's it. For the first monitor (which is not at my current location, so I can't test it right now) and for the TV, I connected directly with HDMI.
As for pushing on the socket, nothing seems to happen when I do that right now. The good news is, at least for the moment, I'm not seeing the problem anymore. Even when hooked up to the monitor, nothing I'm doing now makes it lose signal.
It bothers me that I don't know what the problem was or what caused it to go away, but for now I guess I can just monitor it and see if it stays gone, or if it starts to act up again.