You can be identified as the user of your new phone number by your provider and any law enforcement. Your provider has your location data. Knowing where you live and where you work is most likely a more unique identifier than your combination of first name and last name.
Your old phone provider and your new phone provider also know that there is a lot of communication going on between both numbers. Both can read all your SMS send between the two and have a log of all calls forwarded.
If the person calling you gets your new number depends on the method of forwarding, but usually they wont see your new number. - Test it yourself, by calling from a third phone.
Since your old phone number is already tight to your identity, the only thing you can hide in your scenario is your new phone number. But does that matter, when calling or texting to your old number leads to the same result as calling or texting to your new number?