I would divide approaches to Christian biblical studies by those who doubt the Bible's inerrancy into two categories
- Believers that the Holy Spirit generally inspired the Bible, and Christian doctrine is infallible; but errors may have crept in to the text.
Such people do not actively look for errors, but if confronted by contradictions, scientific errors, moral objections to biblical teaching, theological problems etc., they do not presume that the Bible must be right. Instead, they are likely to turn to church tradition or personal conscience for guidance. Catholics fall into this category, but so do many others. The US Council of Catholic Bishops advises:
The Bible is the story of God's relationship with the people he has
called to himself. It is not intended to be read as history text, a
science book, or a political manifesto. In the Bible, God teaches us
the truths that we need for the sake of our salvation... We read the
Bible within the tradition of the Church to benefit from the holiness
and wisdom of all the faithful.
Many leading Evangelicals also fall into this category. The article Inerrancy and Evangelicals: The Challenge for a New Generation by Nathan A. Finn recounts:
British theologian James Orr... rejected inerrancy outright in his
widely read Revelation and Inspiration (1910)... Southern Baptist
Theological Seminary president E. Y. Mullins, emphasized the
infallibility of the Bible’s message but was reticent, in his
systematic theology (1928), to make claims concerning the text itself.
Beginning in the 1960s, a growing number of evangelical scholars began
departing from their prior commitment to the full trustworthiness of
Scripture. In many cases, the influence of Swiss neo-orthodox
theologian Karl Barth (1886–1968) played a key role in dislodging
plenary verbal inspiration and inerrancy/infallibility.
- Those who affirm the truth of Christianity generally but believe critical study of the Bible enables them to understand better how God has worked with his people over the centuries.
Scholars of this type use the historical-critical method and textual criticism to understand how the Bible arrived at its present configuration. They tend to see God's self-revelation in the Bible as manifesting through fallible human instruments whose particular backgrounds and agendas may have colored their writing. Protestant theologians such as Rudolf Bultmann sought to "demythologize" biblical teaching in an effort to uncover the message of the living God. In Catholic theology, Hans Kung is an example of a scholar who felt that he needed to re-examine church doctrine and tradition in light of the historico-critical method.
The Jesus Seminar included many more contemporary scholars of this type. It sought to discern the actual sayings of Jesus as distinguished from those that might be authentic and those which they saw a clearly from a later period, reflecting the church's teaching rather than what the historical Jesus himself would have said. Its website says: "Among the findings is that, in the judgment of the Jesus Seminar Scholars, about 18 percent of the sayings and 16 percent of the deeds attributed to Jesus in the gospels are authentic."
However, it should not be presumed that all scholars who fall into this category would reach such radical conclusions.
Christians who doubt the Bible's infallibility us various methods to determine which parts are trustworthy and authoritative. Some rely on church tradition and pastors for guidance. Others rely more heavily on their own study. Tools used by Christians in their search for truth in the relation the Bible include books by modern theologians such as Barth, Bultmann, Kung and their followers, source criticism (the historical-critical method), textual criticism, comparison with other historical and philosophical texts, science, common sense, moral philosophy, the history of religions, prayer, conscience and intuition.
I realize as I post this that I have not really taken the OP question by the horns but have described two general Christian approaches to the issue, rather than providing much detail about how they arrive at their conclusions. I will post my answer anyway, because I think it provides useful research on the OP question, as well as a few specifics on methodology. In the meantime, readers may wish to look up individuals mentioned to understand their hermeneutical approaches better.