Although I cannot speak with authority, I see no future problems with you becoming a Catholic.
You do have to be properly baptized. Christian congregations that validly baptize use the Trinitarian formula:
[Name], I baptize you in the Name of the Father, of the Son, and of the Holy Spirit.
Even if the Christian Spiritualist Church didn't baptize you this way, you can be conditionally baptized when you enter the Catholic Church. The priest or pastor bringing you into the Church will be able to discern the validity of your current baptism. [Related posting.]
What I cannot speak to is the validly of your marriage but there is a high likelihood that you will be able to have your current marriage blessed by the Church. Your spouse, assuming he is not a Catholic, would not have to join the Church though he would have to agree to be a witness to your promise to raise your children as Catholics and not to interfere with your efforts in this area.
As to the Petrine Privilege, we have answered several questions dealing with this issue here:
Petrine privilege, also known as the privilege of the faith or favour of the faith, is a justification recognized in Catholic canon law allowing for dissolution by the Pope of a valid natural marriage between a baptized and a non-baptized person, for the sake of the salvation of the soul of someone who is thus enabled to marry in the Church.
I hope this helps,