Sadly, if there are items in question and you two are fighting over them, this is where having the money to take it to court comes in. If you do not have the money to hire an attorney you must decide whether or not it is worth fighting over. This is where that part comes in about “picking your battles”.
If there are children involved, you and your stbx could be in for a very rough time if you do not soon resolve some of the issues. If you are fighting over material things, I can only imagine what the issues with the children will be.
If you have an agreement that does not specifically state what is who’s then you must either agree or take it to court. If the person in the marital home does not wish to “give up” the items in question, and the agreement does not specify it will have to be taken further. If it goes to court I believe that it will be assumed to be marital property and therefore divisible.