From a Christian perspective there is no theologically distinct "Old Testament God", so this question can only really be cogently answered from a Jewish perspective, and as someone who isn't religiously Jewish I don't feel equipped to say.
I mean God’s characterization in the Old Testament
Which parts of the Old Testament? There are anywhere from 39 to 49 books in it.
I'm still waiting for one of our red or yellow avatar Bible experts to answer this very simple question.
It’s just his overall characterization across the entire Old Testamemt
In the J source? The E source? The Deuteronomistic History? The Nevi'im and Ketuvim? Why is this (again, outside a specifically Jewish theological context) a relevant category in a way that a version of the category that includes New Testament writings is not? Are we looking at the neo-Marcionite stereotype of muh Old Testament God, or at the actual Old Testament? Are we admitting midrashim as relevant? Quasi-midrashic interpretations by Christian Old Testament scholars like Walter Brueggemann and Kathy Darr? Both? Neither? Work by Old Testament-focused devotional writers like Rachel Held Evans (pbuh)? What makes this a political question that needs to be in the Individual Politics board? So many questions, so little time!
Are you a Documentary hypothesis fan? I found RJ Alter persuasive against it in The Art of Biblical Narrative, myself.
I was taught the documentary hypothesis as established fact but I wouldn't say I'm a "fan" exactly. I respect Alter very much but haven't read that book; what's his argument?
I should clarify he doesn't go as far as some have in abandoning the documentary hypothesis totally, but he shows where there is a strong internal coherence in theme, language and imagery tying together elements that have been attributed to the different sources. So he shows how the story of Judah and Tamar are closely connected to the Joseph story rather than just an unrelated interpolation, and he challenges the idea that what a modern scholar might see as contradictory duplicate accounts side by side really are so.