To Who?
The State Legislatures involved and/or Joint Session of Congress. SCOTUS, especially with the current makeup, are very cognizant that their decisions do not carry any, lets call it “political” recourse from the people (its a lifetime appointment, not elected). If there is a constitutional procedure that specifies legislative involvement, the SCOTUS will always punt the ball to them to figure out what to do. They may issue guidance when doing so, just like they did in the denial of the Texas case.
Have any new lawsuits been filed by State legislators stating that their specific federal constitutional rights were violated by the state executive branch?
There has been one pending with SCOTUS from PA since October. Docket #20-542.