I'm going to try to keep this short but please forgive me if it gets very long while trying to include enough info to give an accurate picture.
I'm in a live-in relationship with my high school sweetheart. We reconnected on Facebook about a decade ago and both of us left unhappy relationships to be together. Because we joined the kink community early on, I decided we needed to sit down and work out a set of rules that would work for both of us. What we came up with was a monogamish arrangement that would allow either of us to do pretty much whatever we wanted but only if it was openly and honestly discussed first and reasonable demands were met. Things like meeting the other person before anything happens, if they're not already on our list of mutually known and trusted friends. People on our list enjoy very few restrictions.
Things requiring discussion and agreement first ranged from activities as simple as meeting someone for coffee to kinky-play-plus-sex. One of our specific rules was that anything that happens only online is cool. No discussion or disclosure required. This was something I wanted because I'm a furry and have close friends in the furry community, a few of which I engage in cybersex with. What I meant, but didn't specify when we were negotiating our rules, was online-only play was cool with people we'd never met in real life and never would. Not being specific is on me, but I consider it borderline to breaking the spirit of the rule that he was sexting with women from his workplace. It irked me some but I let it go.
Time goes by and it comes out by accident that he met up outside of work with one of these women he'd been texting with. I had borrowed his phone to call for food to be delivered and when I opened it the first thing I saw was a text from this woman about how nice it had been to see him again. I knew the name and that she didn't work with him anymore. At first the story was that he'd met her at the place where she works now, since she left the place where he still works. At that point his story was that nothing happened but talking, and it was only once. That was a violation because I don't know this woman, of course, but if his story was true, it was a minor violation. So I kept my cool and discussed it with a friend so I could try to handle the breach in a calm and reasonable manner. My friend helped chill me out, but automatically assumed he'd cheated on me. I wasn't ready to jump to that conclusion yet, so I talked it over with him, reminded him that he broke the rules and needs to do better.
During this discussion he got defensive and it came out that he'd met up with her twice. Again he claimed nothing happened. That was too far and I blew up at him. I know I shouldn't have, but damnit I was mad at the betrayal of my trust. Things calmed down and I told him if he cheated just tell me now and I'd forgive him this one time, but he'd have to do things right in the future. He insisted it was only talk.
Life went on but the situation kept eating at me, so finally I snooped and saw in the text exchange that one of those times he'd visited, he'd taken her a cup of coffee. Of course this made no sense, so I asked him why he had to take her coffee if they met at the restaurant where she worked, which should definitely have coffee available. He admitted he went to her house, which was why he took her coffee. Again he insisted that nothing happened. I like to think I'm a trusting person, Dan, but I'm not a fool. At this point there is no way for him to convince me that he didn't cheat.
Really angry now I snooped further and discovered there was at least one other woman he'd met up with outside of work!
I know this looks like a DTMFA situation but I'm stuck due to serious health issues that prevent me from working and I don't have options. Just to answer a question I expect you to ask, the issues do sometimes prevent me from being able to have sex for a week or two at a stretch, but no longer than that. So I have to stay. But I want to know if I can salvage any part of what I thought had been a good relationship. Or do I just stop caring what he does and get my needs for emotional support and affection met elsewhere? Also is there any reasonable argument for why someone who broke rules and continued to lie about it even after an offer of clemency might not have cheated? I don't think so, but maybe you can see something I don't.
Facing Uncertainty Realistically
I had to go back to the start of your letter to reread the rules for fucking/sexting other people so I could track what exactly your boyfriend did wrong.
If I follow you, FUR, your boyfriend sexted with people he knew IRL and he may have—okay, okay: he did—have sex with someone else, and possibly two someone elses. In both cases he was sexting with these women before he
may or may not have slept with them. And while he's allowed to sleep with other women—while you're both allowed to do "pretty much whatever" you want—he's supposed to discuss new partners with you in advance and meet whatever reasonable demands you might have... unless we're talking about someone on that pre-approved list of known and trusted friends, in which case anything goes.
So he can sext and fuck other people. But he's not supposed to sext with people he knows IRL (a rule that had to be amended/clarified after it was adopted) and he's supposed to clear unlisted partners with you first.... and you would dump him if you could dump him but you can't dump him because you have health issues that prevent you from working and thus are financially dependent on your boyfriend.
First, about those health issues: the fact that you can't have sex for a week or two at a time doesn't give your boyfriend carte blanche to fuck other people in contravention of the rules—rules that allow him to fuck a number of other people pretty much however, wherever, and whenever he wants. His options weren't limited to you or nothing for two long, miserable, sexless weeks (oh, the humanity). If he was desperately horny and you were out of commission, he could've rung up someone on the pre-approved list—or presumably cleared one of these two women with you first.
You were willing to forgive him for the sexting a person he knows IRL infraction. So I imagine that if he'd come to you and said, "I was sexting with someone I met at work and that wasn't right and I'm sorry. I'd like to get your forgiveness for that before asking for your permission to fuck that person," you would've been inclined to give him/them your blessing.
But he didn't do that. He broke the rules and now you want to dump him. But you can't. Because you're dependent on him for food and shelter and have no other options—no family to fall back on? no friends who can take you in?—so you'll either have stay and hate him or stay and forgive him... for sleeping with other people. Which he's allowed to do. But this isn't okay because he slept with the wrong other people.
Rules are important, of course, and every open relationship has its own set of rules. If your boyfriend was chafing at the rules you two agreed to, he could've reopened negotiations and hammered out a new set of rules. But that wouldn't work if breaking the rules was his goal—if transgression was his goal. Which it sometimes is, FUR, and if this was about breaking the rules... well, that would explain why your boyfriend continued to lie even after your offer of clemency.
So what now?
You can stay and live together and ice your boyfriend out and risk being dumped by him—the "stop caring and get all my needs met elsewhere" option—or you can forgive your boyfriend for fucking (the wrong) other women.
Some suggested reading if you decide to go with option #2: The State of Affairs: Rethinking Infidelity by Esther Perel.