Let's think about regular linear regression, and to make it concrete, let's say we are trying to predict height of people. When you regress heights against just an intercept term and no predictors, the intercept term will be be the height averaged over all the people in your sample. Lets call this term $\beta_0^{\text{no predictor}}$
Now, we want to add a predictor for sex, so we create and indicator variable that takes a 0 when the sampled person is male and 1 when the person is a female. When we regress against this model, we will get an estimates for an intercept term, $\beta_0^{\text{male reference}}$ and coefficent of the sex variable $\beta_1^{\text{male reference}}$. The estimated intercept is no longer the average height of everybody, but the average height of males, the coefficient of the sex variable is the difference in the average height between males and females.
Consider if we decided to code our indicator variable differently, so that the sex variable took the value 0 if the person was a female and 1 if the person was a male, in this specification of the model we get the estimates of the intercept and coefficient $\beta_0^{\text{female reference}}, \beta_1^{\text{female reference}}$. Now $\beta_0^{\text{female reference}}$, the intercept term, is the average height of females, and the coefficient is the difference in average height between females and males. So
$$
\begin{align}
\beta_1^{\text{male reference}} &= -\beta_1^{\text{female reference}}\\
\beta_0^{\text{male reference}} + \beta_1^{\text{male reference}} &= \beta_0^{\text{female reference}}\\
\beta_0^{\text{female reference}} + \beta_1^{\text{female reference}} &= \beta_0^{\text{male reference}}
\end{align}
$$
So, by changing how we coded the indicator variable we changed both the value of the intercept term the coefficient term, and this is exactly what we should want. When we have a multivalue indicator, you will see the same kinds of changes as you specify difference reference levels, i.e. when the indicators take on the value of 0.
In the binary indicator case the p-value of the $\beta_1$ term should not change depending on how we code, but in the multivalue indicator case it will, because p-value is a function of the size of the effect, and the average differences between groups and a reference group will likely change dependent upon the reference group. For example, we have three groups, babies, teenagers, and adults, the average height difference between adults and teenagers will be smaller than between adults and babies, and so the p-value for the coefficient for the indicator of being an adult versus a teenager should be greater than an indicator of being an adult versus a baby.
Among the many possible ways to analyze this dataset, a one that I would try first is multivariate linear regression. Multivariate means that you have multiple outcome variables (the six Likert scales). Linear, rather than logistic, means that you treat each Likert scale as a continuous output. (It's true that ordinal logistic methods are more appropriate theoretically for rating scales, but that seems like overkill for a first attempt.) Rencher, "Methods of Multivariate Analysis" seems like a decent introduction.
Best Answer
Further information given in comments by the OP suggest that the problem here is separation or quasi-separation since 85+% of the cells formed by a complete cross-classification are zeroes.
To answer the original question posed first: the finding of collinearity in the model is not necessarily a red flag as it is sometimes treated. It may not even be an orange alert either. It is conveying important information about the data-set which needs to be looked at before further interpretation is made. This task would certainly need to be undertaken by someone knowing the scientific question and the background to the data-set, information which we do not have.
Separation is a topic which has been handed elsewhere on this site and fortunately there is an excellent answer in this Q&A How to deal with perfect separation in logistic regression? (in my opinion the highest voted answer, not the accepted one is the one to go for if you are short of time to read them all).