In logistic regression, logit Y = ax1+ bx2 +cx3. a variable has a coefficient attached to it. So we can directly measure the change in y per unit change in x. Can I get a similar relationship from odds ratio? per unit change in x contributing to how much change in y. Basically I want to understand the contribution to score. E.g. a variable has odds of 20:1 compared to 6:1
Your first line is incorrect. In a logistic regression, Y is not linear in the parameters. Indeed, that's one reason why we need logistic regression. The logit of y is linear in the parameters.
The parameters output from a logistic regression lack an intuitive explanation, unless your intuition about equations with exponentials in both numerator and denominator are a whole lot better than mine.
The odds ratio does what you think the parameter does. That is, the OR measures how the odds of Y change per unit increase in each X.