You have a multitude of issues, here. You are supposed to get notice of all probate proceedings. But how to respond appropriately is something you need a lawyer for. Ultimately, the decision will be made by the judge, but it really sounds like this situation is ripe for additional conflict beyond the immediate issues. It may be best if the judge appoints a public administrator, as you suggest.
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Unfortunately I think you would be well served hiring an attorney if there is conflict among the family. You being named as next in line, you can either serve with the guidance of an attorney or decline on the agreement of all parties that a disinterested third party will step in. It will be difficult for you to negotiate all of this on your own, and probably will cost the estate more in the long run. It sounds like there are already issues that are causing the current executrix to be removed or step down, so it is best that everyone get on the same page to reduce the chances of additional issues going forward.
You need to file a motion in probate court to replace the executrix of the probate and in his or her place appoint someone of your preference. If you don’t want to serve in the capacity, and you want an outside person then you should choose a qualified professional fiduciary with credentials for the court’s consideration. You are also going to have to provide a good factual basis for why the executrix needs to be replaced.
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