So, you have crude OR value for a given variable and the "adjusted" OR value for the same variable when combined with other independent variables in a logistic regression model, and:
1. The adjusted OR for the target IV is larger than the crude OR for the IV; and
2. No IV in the full model has a regression coefficient different from zero?
That tells me you can not assert the adjusted OR is different from zero, so you're not really able to make a meaningful comparison with the crude/unadjusted OR for that variable.
Assuming the adjusted vs. crude comparison was true, what could be going on?
1. Your sample size might be too small, relative to the number of IVs in the full model, to afford much statistical power to detect that an estimated LR coefficient is different from zero.
2. There may be one or more suppressor variables among the other IVs in the full model.
3. You didn't indicate whether you included interaction terms. If so, it's possible that there are some moderating influences among the IVs.