I can relate wholeheartedly with what you're going through, I don't have OCD but rather ADHD. I'm 37 now and I always knew something was wrong with me, but my mom was in complete denial about it (probably embarrassed about it in my opinion), she is still in denial about it even after I was diagnosed by a psychiatrist! My grades were good, I was good at sports, a good musician, good writer, so no teachers ever helped (my mom was the elementary art teacher in the same school district

). I compensated for the disorder for literally decades until I was diagnosed with cervical cancer, then I just decided to take control of my own health. Long overdue.
My problem is that I have trouble paying attention to detail, any details at all, and as a result I make stupid mistakes, forget things (I have the long-term memory of an elephant but absolutely zero short-term memory), and on the flip side I get so worked up when things do go wrong or something happens that I have trouble calming down. I won't sleep for days, I don't eat, all I do is worry about it and eventually I'll become nauseous and I'll throw up. I had a complete hysterectomy when I was 32 and they took both ovaries, so I immediately went into post-surgical menopause, and the lack of estrogen, progesterone, and testosterone makes ADHD a lot worse. Mostly the attention part, thank God because I couldn't take getting any more worked up about things. I'm medicated (helped me so much, which makes no sense since I take stimulants and they calm me down and allow me to sleep, go figure) and I go every week to counseling and to group therapy twice a month. I don't know what's worse, the early menopause (hormone replacement therapy only helps so much), the ADHD, or the disease of addiction, which is much harder to deal with when it's influenced by the ADHD. Somehow I'm doing well, I've been on the upswing for almost 2 years now, but it's a daily struggle. Today was a good day so that's about all I can control until I wake up tomorrow.
My pets are a godsend and the only reason I got through the cancer. When I thought I was going to lose my house because I couldn't work for an extended period of time, I thought I was going to lose my dogs, my birds, and have no one. I was in bad shape while going through that, and a lot of people didn't understand why I was so concerned about my pets when I needed to be concerned about myself. That made me angry every time it came up. Pets are used as emotional support animals for a reason.
I had two of the episodes just like you had, one with my dog just after I got sick and the other with my green cheek conure about 6 months ago. With my pug I came home from work around 8:30, I had worked from 2:00 to 8:00 that day and had a late customer. It was hard to concentrate at work at that point, all I could think about was that I was going to lose everything I had, lose my pets, lose my house, become horribly sick, suffer, then die. And I worried and worried and worried about it. So I came home, walked in the front door, and my cattle dog greeted me at the door like always, my pug was asleep on the couch like always. I walked into the kitchen and yelled for him to come get his dinner, no response. I said hello to my Quaker parrot, then my budgies, and yelled again for my pug to come eat his dinner, and no response. I got a chill that went over my entire body. He was 13 at the time so I freaked out. I ran into the living room and I actually shook him and he still didn't move! I screamed "Jaeger! Oh no, not now Jaeger!" and I just started balling my eyes out as I fell to my knees. Then he lifted his head up and yawned, then looked at me like "Why did you wake me up?" He was in such a deep sleep I was sure he was dead, I yelled 3 times for him and literally shook him! I was a wreck that night and the next day I was afraid to go home. I didn't want to open the door and find him like that again. It was a struggle for the next week or so, I couldn't sleep, when I did I dreamt that he had actually died, it was all I thought about all day at work, and when I would forget about it for a while, the minute I remembered I would get a chill and something like an electric shock down my body. And I was afraid to go home. I finally did get over it or at least I stopped being consumed by it, but it took time. I think it brought me much closer to my pug in the last year of his life, but wouldn't you know that when he died a year later at almost 15 years old, I found him dead on the couch when I came home from work one night. I shook him and shook him, but this time he was stiff and cold. I was expecting him to pass away soon, he was old, but you're never prepared. But having him die that way and finding him that way made it a little less traumatic, believe it or not. I had obsessed over it so much that when it did actually happen exactly the same way it was like I had already experienced that moment and it lessened the blow.
Everyone is different and I can't tell you what will help you, but just know that what you're feeling isn't irrational at all, you love your bird very much, he's your family, and the way that you're reacting to the thought of losing him is completely normal. It's a rational fear, not only because you love your bird so much and the thought of losing him is horrifyingly painful, but because it's something that is out of your control. What you can do is tell yourself "He's here now, he's fine, he's healthy, and he loves me". And what you can control is making the most of your life with him every day.
"Dance like nobody's watching..."