Welcome to the forum, and to the Church!
I'm sorry this is such a difficult experience for you. I hope that it goes well for you.
Have you been completely open with your priest about the extent of your fears? He may not understand the magnitude. Maybe you can work something out with him, like he puts his and on your shoulder instead of your head, doesn't apply too much pressure but just guides you, submerges as briefly as possible and mostly splashes over head,. etc.?
If you must suffer through it, you are in good company. Do not feel that it makes your Baptism any less if it is not a joyful experience to you. Think of the martyrs, many of whom were baptised in their own blood. Baptism is a mysterious participation in the death and resurrection of Christ. If for you, it really feels like death, this is a crown that you will be rewarded for.
Baptism must be an act of the community. It is not a magical act of the priest. There should be no such thing as a private baptism, or a private Eucharist. You are becoming a member of the Body of Christ, and it is a logical contradiction for this to be done apart from the Body. Being received into the congregation is a part of it. However, the actual act of Baptism should be private, with just you, the priest, and a female servant, such as his wife, in order to protect your modesty. Then you emerge from the baptistry to join the faithful for the Liturgy of the Eucharist, and to commune with them for the first time. At the end there is a procession around the Church, as the angels in heaven, and the people in the Church, celebrate over your repentance and rebirth. In Catholicism, religion can be private, with private Masses, and a private spiritual life apart from the community. This is not possible in Orthodoxy. The assembly of the faithful is central. However, if there is pressing psychological need to make accommodations to make the day bearable to you, this would be at the discretion of your priest (or his bishop if he needs to take advise on how to deal with it), so you should discuss it openly with him.
You have mentioned a cultural barrier. Perhaps it is the culture of the priest to just say don't worry, it's ok, and to present a solution, and if you don't protest he thinks you've accepted and are happy. He may be expecting you to be very forceful with your concerns, that may be what he is used to and he may not appreciate how concerned you are apart from that. Don't be afraid to press, to insist that it be discussed more. Just tell him how uncomfortable you are, and that you need to talk about it. He is becoming your spiritual father, and it is very important that you develop a relationship where you can be open with him, even if it requirers cultural accommodation on both of your parts to get there. If you need a practice run, just tell him, "Abouna, I need to practise this and see if I can keep my fear under control, please help me with this".