Unfortunately, Genex (www.swabtest.com) is not a testing company I am familiar with. They seem to have their own custom tests and custom results without giving you the ability to compare yourself against all the other people in the test base as you can do with the standard testing companies.
If Genex allows you to download your raw data, you may be able to upload that to GEDmatch and then compare yourself to your sibling. GEDmatch would show you where you and your sibling half match (one of your chromosomes of any pair) or fully match (on both chromosomes of any pair). One of each pair is from your father and the other is from your mother.
If you are full siblings (you have the same father and the same mother), then you will share with your sibling about 1/2 of your father's DNA on the chromosome from your father and 1/2 of your mother's DNA on the chromosome from your mother. That means that the pair of chromosomes from both parents will match on 1 chromosome half the time, both chromosomes 1/4 of the time and neither chromosome 1/4 of the time. Note that 1 chromosome x 1/2 + 2 chromosomes * 1/4 + 0 chromosomes * 1/4 = 100% which is what Genex is reporting to you.
Now, if you are half siblings with a mother in common but different fathers who are not related to each other, then you will share on average 1/2 of the DNA from your mother but zero from your father. Here 1 x 1/2 + 2 * 0 + 0 * 0 = 50% which is what I would assume Genex should give half-siblings with unrelated fathers.
Your mention of double cousins is not applicable, since you are certain you have one parent in common, and double cousins have no parents in common.
But if you are half siblings with a mother in common but different fathers and the fathers are brothers to each other, then you will share on average 1/2 of the DNA from your mother. Your father will share half the DNA of his brother. And you and your half sibling should have about 1/4 of your DNA from your fathers in common. If you could look at this in GEDmatch, you and your half-sibling would match on one chromosome 1/2 of the time and on both chromosomes 1/8 of the time. Here 1 x 1/2 + 2 * 1/8 + 0 * 3/8 = 75% which is what I would assume Genex should give you if you are half-siblings with a common mother and fathers who are brothers.
But because I am unfamiliar with Genex, I would hesitate to say that I am interpreting their results correctly.
For an issue that obviously would be of such monumental importance to you, I would not trust a genealogy Q&A site to give you your answer. I would recommend you spend the money and hire a professional genetic genealogist to give you advice of how to proceed and then do the analysis and confirm the definitive answer that you are looking for.