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.
There's no right or wrong answer here, you can use either one in this situation.
If you say
"I've been waiting for you for 5 hours." you're concerned with right now,
If you say "I was waiting for you for 5 hours.", you're concerned with the past.
Best Answer
Let us be clear about time phrases beginning with "SINCE" and "FOR".
SINCE/FROM refers to POINT OF TIME (10 a.m./year1940/Monday last)
FOR refers to PERIOD OF TIME (10 hours/ 3 days/ 4 months)
They have nothing to do with tenses. However, they assume special importance as 'perfect continuous' form of tenses specially demands mention of time factor.
" Since/for/ from" can safely be used in any other tenses if their presence can by no means disturb the basic concepts of time element of tenses in use.
As regards Present Perfect Tense, it is not a self contained tense; as if it expects some thing else of undefined nature. (e.g., I have sat for the exam.--just completed action--expecting may be result, may be outcome, may be any thing else) We may use since and from here as well.