Since you are making a distinction about public vs. private unions it appears that this is not a constitutional issue. It seems to be one where you don’t like that they generally endorse and support one party over the other. There are very few entities that are neutral. All sufficiently large entities become political to some degree.
This is completely false. People absolutely are forced to join unions.
Furthermore, the overwhelming majority of union members - over 90% - never voted in a certification or authorization election. That means that, even if they're members of the union, their membership cannot in se be considered an endorsement of the union by the bargaining unit. Because it's all-but-impossible to successfully decertify a union in practice[0], it's not uncommon to have the majority of employees oppose union membership, but still be compelled into membership.
> They are, for the time being, in some states required to pay for fair share dues to cover the costs of collective bargaining, grievances, and other administrative costs
People within the bargaining unit are usually forced to pay dues whether or not they are members.
[0] The NLRB has the power to overturn the results of decertification, and a very strong incentive to do so. In addition, unions have learned that they can change the definition of the bargaining unit after an election is held in order to invalidate the election retroactively.
Why are you asking me? Go find some conservatives and ask them.
Perhaps asking someone who has actually made both those statements would be more helpful than asking me.
This is absolutely not true in the least. Unions tend to support candidates that will help and support them, just like anyone else who backs any other candidate.
Private entities can limit free speech all they want; public entities cannot.