Yes, it is possible to use "ever since" with the present perfect. "Ever since" links two events, a cause event and something that is true from that point in time onward. This is actually ideal for the present perfect tense because one of the uses of the present perfect tense is to indicate that something which began in the past is still relevant in the present. An example of this use is your sentence:
"I always thought it was wrong, but ever since I read it, I've been more confused."
Using the present perfect with "ever since" shows that the speaker has been and is still confused by something which they read in the past.
With the past simple tense, as in your last sentence, it would be more appropriate to use "once" instead of "ever since". This is because "once" links a cause event with something that changed, but does not necessarily link to the present.
Once I did that, it stopped bothering me.
This sentence shows that the speaker was not bothered by "it" as soon as they did "that". It however gives not indication of whether this continues to be true in the present.
I saw your edit, if I understand right you meant to ask if it is possible to use "ever since" and present perfect without past simple, which would look something like:
"ever since" + present perfect + present
Ever since I have been pregnant I feel sick each morning.
The answer is no, this is not natural English. This structure may be encountered in informal English, but it is more natural to have the structure:
"ever since" + past simple + present perfect
Ever since I got pregnant I have felt sick each morning.
Usually, we don't use "since" in the simple past. However, there are a few specific cases in which we can use it as follows:
1- We can use it as a time adverb such as it's long/a long time since I met him,
it's two weeks since I joined the bank, etc.
2- We can also use it as a conjunction where it means "because" such as since you
don't have money, I'll not take you to the market, you should take Mr Adam's
permission since he is our General Manager.
As for the sentence "Deputy chief constable fights against crime since he was first appointed", I don't think this is acceptable although some people contend that when you have been doing an activity regularly for a long time, you can use since in the present simple for this activity. I don't think it's valid justification.As a matter of fact, the right sentence is "Deputy chief constable has been fighting against crime since he was first appointed".
Best Answer
"I am" doesn't work because it refers to an action that happens now, where as "Ever since I was a child" refers to a whole period of time, most of which is in the past. "I have been" refers to a continuous action over a period of time that began in the past and continues to the present, so that's what you want to use here.