I can relate, but in an opposite way! I think I'm more in his position. I have a husband and a boyfriend. Both of them complain about the same thing "When you're with me, you're texting him too much"
Ok... well, that's fine and dandy. The kicker? In the same moment I get the "You don't text me enough when you're away from me"
Ok. So. My solution: apologize. I'm constantly saying "I'm sorry, let me wrap this up and then we'll ____" whatever it is. So, I have some 'rules' per say. If I know I'll be out to dinner, I text the other "Hey, headed to dinner, going to be MIA for a while" and then during dinner, I try really hard to steal a moment to text "<3" or some kind of hey I love you message. I try REALLY HARD NOT to be attached to my phone. However, it's a level of reassurance that I can FEEL vibrating off of the unattended lover, and I CANNOT ignore that need.
How could I ignore someone saying "Hey, I'm feeling left out, I miss you" ???!!! There's no way!
But yes, I see how you're feeling! I haven't had to experience that side too often. For a while my boyfriend was / I guess still is / seeing someone else. He wouldn't always warn me when he was about to "drop off" so to speak, and I would be HORRIFICALLY offended if he didn't respond. I convinced myself he didn't love me. I convinced myself he was off making a life with this other woman! Crazy, right?!
Maybe share this story with him? Explain that if you went out and didn't say "hi" every now and again, he'd be hurt (especially if he was sitting at home alone!).
As for moving in being a "solution".... it might be! I'm very content crawling into bed with either option, but they are definitely much more content when I'm there next to them.
Poly is a lot of things, forcing yourself to accept an uncomfortable situation, should not be one of them.