The key point you mention is that you and your sister share both half identical and completely identical regions of your DNA. Only full siblings will share significant amounts of completely identical regions. Half siblings will not. The chromosome browser in 23andMe will distinguish these for you.
Full siblings share about 50% of their father's DNA and 50% of their mothers DNA. Considering overlaps, that would on average amount to this:
- 25% that you both share with your father only (half identical)
- 25% that you both share with your mother only (half identical)
- 25% that you both share with both parents (completely identical)
- 25% that neither of you share with either parent.
As an example, here is what the chromosome browser at 23andMe looks like for a set of full siblings:
The lighter color are the half identical matches, and the darker are the completely identical matches.
What 23andMe is reporting in this example as "Half identical 2778 cM" includes the 845 cM completely identical segments. If you are full siblings, you should see something similar to this, where about 3/4 of the chromosomes are either half or completely identical, and 1/3 of the matches are completely identical.
Unfortunately, I can't show you a comparison of what the chromosome browser would look like with a half-sibling match (or an uncle/aunt to nephew/niece match which would look the same) because I don't have any in my own results. But the chromosome browser for one of those would fill about half the chromosome space (compared to 3/4 filled for full sibling, as above) and almost all the colored lines would be the lighter color (compared to 1/3 of the colored lines being the darker color, as above).
Looking at your own chromosome browser and using the above information, especially with regards to completely matching segments, you should see clearly whether the browser is indicating a full sibling relationship or a half sibling relationship.
The other important check you can do is to look at the people you match to, and the people your sister match to. If you are full sisters, then you should both match to close relatives (2nd cousins and closer) on both your mother's side and your father's side. If you have different fathers, then you will only share close relatives on your mother's side but will not share close relatives on your father's side.
All the above assumes that if there are two fathers, then they are not closely related. If they are, say they are brothers to each other, then you've got something in-between, and you will have some completely identical matches, about half as much as if it was the same father. And you will then share close relatives on both sides.
If you do not clearly show as half-sisters, you may want to get a professional genetic genealogist to help you sort this out.