As an initial observation, if the decedent lived in Tennessee, then I would assume that the probate court exercising jurisdiction over her estate is also located in Tennessee. Even though you live in Nevada, you will need to take action in the same probate court that assumed jurisdiction over your cousin's estate.
I suspect that you will need to hire a local probate attorney in Tennessee to assist with this.
The process varies from jurisdiction to jurisdiction, but typically there is some sort of process whereby you can either file a motion or a petition with the probate court challenging the standard intestate succession (intestate succession is the formula used by the court to determine who receives the possessions of a person dying without a will).
However, keep in mind that time is a factor. If you sit on this too long, you could lose your rights to intervene in the probate process. Depending on local rules, you usually have a window of time from the date that probate opens to intervene and state your case.
Tough situation. I am curious, why do yo say that the heirlooms that belong to your side of the family? There may have been some spoken agreement or understanding between the parties about it but courts really struggle with that because of proof problems. If these are the type of assets where possession determines ownership, then your cousin should have had a will that said they were yours.
Does any family member have a relationship with the dad. Maybe they can come to some accommodation.
You need to discuss this with a Tennessee lawyer as the decedent was apparently a resident of Tennessee. I suspect the Tennessee lawyer will tell you that the property goes to whoever gets it under the laws of Tennessee when someone dies without a will. But do check with a Tennessee lawyer.