I think the alternatives proposed imply that there is a specific, accessible truth against which your less nailed-down truths could be measured--but what I understand from your question is that independent standards are not very available if they exist at all. After all, if they did, you could simply contrast the "truths?" with those, well, truths.
For that reason I think I like the "degree of truth" phrasing. Yes, it sounds strange, and that is a good thing. It brings more attention to the sense of hesitation you want to convey.
In mathematics, we refer to "degrees of freedom"--the lower the number, the more constrained the quantity you are referring to. Example: if you know you have 10 numbers which add up to 17, and you know what 9 of those numbers are, you have 0 degrees of freedom for the 10th number--it is whatever, added to the known 9, equals 17. If you have 2 unknown numbers and 8 known, you have 1 degree of freedom (freedom to chose a value for one of the two numbers, which immediately constrains the remaining on--people use this concept, whether or not they know it, when completing Sudoko puzzles). "Degree of truth" evokes the mathematical "degrees of freedom" notion for me, and I entirely approve of it.
On the other hand, I am much less creative in my own use of language than I wish I were.