When traveling to a new Schengen country could I switch to the US one to essentially restart my 90 days on that passport?
There is no way to "switch" when traveling between Schengen countries. The only way is to leave and re-enter the Schengen area. But switching doesn't give you another 90 days because the 90 days applies to individuals. Otherwise, anyone who was a dual national of two third countries could live in the Schengen area without a residence permit.
I have won a long standing case which for which I have the letter stating I’m citizen from the courts as well as a birth certificate.
That should be enough to keep you out of trouble in most cases, possibly in connection with your third-country passport and an explanation that you have not yet received the Italian passport. It might be helpful to have a pending application for the Italian passport but most authorities will not pursue the matter that far. Legally speaking, you are Italian; administrative fines could be levied for failing to register, but you can't be deported for overstaying, and the administrative fines must be similar to the fines imposed on that country's own citizens for failing to register.
In anyone’s experience would either of these possibilities get me out of any issues with overstaying the 90 (it’s only by a week or so!)
It depends on the country, on your movements, and on the specific issue you're trying to get out of -- in other words, on the reason you have come to the attention of the authorities. If you're leaving the Schengen area, showing the letter and birth certificate along with your third-country passport is very likely to eliminate any trouble, though there may of course be some delay as the officers evaluate the documents.
If you have already applied for an Italian passport, the simplest course of action is probably to stay where you are until you receive it, if possible. If you can't, you are unlikely to run into much trouble anyway. If you've been in the same country for more than three months it might be a good idea to go to another one, at least for a few days or weeks, to avoid any registration requirement.
(As WGroleau points out in a comment, even if passports for different countries were counted separately, you'd still need to ensure that you were outside the Schengen area at midnight when you switch, because if you were to leave and re-enter using different passports on the same day, that day would have to count against both 90-day periods. Day 1 on the second passport would be day 90 on the first. When you want to switch back to the first, on day 90 under the second passport, that's day 179 on the first passport, for which you have already used up your 90 days, so you would have to wait two days before you could legally be present under that passport.)
