OP, I do feel for you here.
You are only 23 and the brain hasn't even matured fully until around 25 years old. And that's to remember, as well, that you haven't probably had the benefit of time for general life lessons ( apart from having a baby, of course).
Add in, that at 23, your formative adolescent years were, also probably, formed around social media and the assumption that all one reads on it can be believed. Not many schools teach critical thinking ( in a robust way, anyway).
As a PP quoted, on another thread Ronald Reagan said "Trust, and Verify". Verify means to use reputable sources. I was no fan of RR but the message is clear...look at BBC Verify, we have to be sure that what we read, and then choose to believe, is from robust, verifiable, reputable sources, not TikTok...this type of SM FEEDS your anxiety, it like a little gremlin in your shoulder, laughing, cackling at your spiralling distress. This is not want you want, is it?
If you truly want to tackle your anxiety, you need to actively come off SM for situations like this, I do wonder, whether there is part of you that feeds off the misinformation and the anxiety that it then causes?
It's doing you and baby harm.
Having said that, you appear to feel reasurred and able to go ahead and get your baby immunised. But this type of scenario will crop up again, and it will be tempting to feed off absurd, cruel misinformation again. This is how you could try and understand the nature of this, as being absurd and cruel.
The human psyche is programmed to dwell on bad news, sad stories, gossip...as PP have said, there are very few threads on TikTok that are pages long on how they immunised their baby, and it was all fine, is there?
I think the very first step for you in addressing your anxiety is to get off SM.
All the best, OP. By the way, a reaction to the vaccines is generally seen as a good thing, it means your little one's body is working to protect them against these diseases. ( MMR, typically produces a reaction 10-14 days post vaccination). Whilst it's never nice to see your child unwell ( especially if you're a lone parent), it is normal to have a reaction, typically, a bit grumpy, ( they may be achey), a temperature, maybe off their food, possibly upset tummy. Keep Calpol and Ibuprofen syrup to hand, weather the storm and know you're doing the right thing.