Use the "one pushup" technique.
Any time you find yourself thinking and/or feeling the kind of concern you're expressing here, self-consciousness about your 'dad bod', worries about how little you've been doing about it...
take a break and do one push up. Just one, it's okay. You won't break out in a sweat. Form is key, so if you can't fully do one that's okay. Try modifying it with your knees until you can knock one out.
That's it. Do one pushup in direct response to your brain thinking and body feeling these things. Every time you can possibly manage, respond to thought/feeling with a specific action.
There are a few tricks to this. First and probably most obviously, is that it will/might/hopefully will give you enough inertia to maybe take it past just the one pushup, to two. Or one more than you did yesterday. Or, while you're down there, some stretching. Just the one pushup is okay, but it's also okay to do more.
Also this gets you in the habit of using your worry and self-consciousness as fuel for motivation. It directly converts negative, stressful thought patterns into physically healthy actions. If you repeat it sufficiently, it might start to remap your brain in a way where exercise does become as much a part of your daily routine as worrying is now.