Worried about being fired

I've been at a job for about 3 weeks as a Manager and am worried about being fired. So far numerous people have quit and been fired. I've never quite dealt with a working environment that openly spoke about getting rid of other employees and with these type of politics plus such a loosely defined role. I'm a paranoid person but frankly feel like the paranoia is justified given the working environment. I'm starting to get weird vibes from my boss, ( doesn't look me in the eye, ad hoc projects i was in meetings for, I'm no longer in. Boss claims he doesn't want to overwhelm me too quickly but who knows.) Recently a job was posted with a similar sounding title to my own. There's been so many people that have come and went that it's difficult to say if it's for my position or to backfill. I spoke with the boss but it hasn't been that long and doesnt want to ramp up too fast. What's difficult to read is if the boss would be straightforward with me or pull the rug. I'm sure most people would say this in my position, but I truly feel that I can provide immense value to this company. If I were to be fired it would be because of perception/politics. My plan is to be as proactive as possible and to manage up. I just hope it's not too late.

Am I being paranoid, or should i start interviewing with the other companies to hedge? Curious to hear advice on how to handle this situation. Thanks.