The feeling of relief/calmness is almost immediate, and likely a result of getting past the pre HRT anxiety you are describing.
The mental stuff seems to happen rather quickly for many of us... but give it a month or so, and you will likely see the difference between a happy brain because you are doing something for yourself and a happy brain because it is on the right fuel.
And things are not always happy. Life does not work that way. They just tend to default to something more positive rather than dysphoric. You will not be immune to dysphoria, but you may find HRT provides you some strong armor.
As for method of administration, that may have something to do with your doc and perhaps your insurance. You can always ask your doctor questions.
I am on injections.
Pills go through the liver but they work fine. Everybody's liver is different though, so it may take a bit of trial and error to find the proper dose.
Other methods bypass the liver so it is a little easier to find your proper dosage quickly for some people.
Most doctors know one HRT method better than the others so I do suggest working with the treatment they suggest, at least to start, and see how it goes. Why be a guinea pig for your doctor if they have already treated other trans patients successfully?
IME, people here tend to have their favorite form of HRT, and it tends to be the one they are currently on.