HMRC Name Change

I recently changed my name via deed poll and need to update my records with HMRC. However, its impossible to login to my personal tax account. The access code is being sent to an old phone number I no longer have access to and I can't change how I receive the code because I don't have a valid passport. It feels silly to renew my passport from a kid in my deadname, when I will have to then pay again to get my name changed on it.

I have read online that if I am PAYE (which I am), I can simply update my employer which will automatically update to HMRC. Is this true? In addition, if I do update with my employer and HMRC can't change it, will I then be unable to get paid?

Many thanks for any help xx