I don't think self-soothing exists at this age? eyes 4 YEAR old DD who is still entirely unable to self-soothe
Seriously OP it's just that particular baby's personality/needs and I think it gets harder if you try to resist or correct it. I totally get that you have to attend to other children so obivously try not to feel remotely guilty if you need to put the baby down so you can get on with basic stuff. But I honestly wouldn't be putting her down (when she's crying or unsettled) DELIBERATELY at this point in her life.
I know a lot of babies who never needed that level of closeness/comfort and some (admittedly fewer) who just did.
Realistically you have a life to get on with but I think you will just add ot your own (and obv your baby's) stress levels if you start deliberately putting her down/leaving her howling just because you think you SHOULD be/someone tells you that you should.
I'm not the best giver of advice, perhaps, because my Velcro baby is now a Velcro 4 year old and as I say above, still rubbish/incapable of calming herself. But she's still tiny, at 4, in my opinion. 4 months is tinier still. They will get there. They also turn out to be extraodrinarily loving toddlers/small kids, still keen on cuddles long after the majority and capable of incredible acts of kindness/sensitivity of their own.
I knwo it probably doesn't help but the medium/long game is worth playing.
Plus as I say I honestly think you will just stress yourself out even more trying to fight this unwinnable/pointless battle.
Dont' feel bad that you need to put her down. She will be FINE. But don't try to teach her to 'self-soothe' before she is biologically/temperamentally incapable. I think it would make her more insecure in the longer term.
Good luck, I do know how wearing it is to have a baby who just CRAVES you at all hours of the day and night.