I hate to be a sucker for my own state, but Rhode Island could easily fit the bill, with some tweaks. The reason for that is that "parliamentary language" - the language more conducive to a Parliamentary government - was never really removed from the State Constitution, which was descended from the old Royal Charter. So it could work. However, the problem with Rhode Island's implementation was that if you tried applying separation of powers to it (which the current Governor has tried), things can get pretty messy - other than the fact that the current system is as much of a pork/corruption-filled machine as one can get. Still, it could work.
Now, for me, I would prefer to improve on it to make it really work, virtually all of this post-POD. So what could happen is the following:
*Within the General Assembly, over time, the House gains more power than the Senate - that is relatively easy since the trend is pretty much in that direction anyway in OTL
*The Governor becomes an essentially ceremonial figure, though he does wield lots of power
*In the 19th century, during those periods when there was a backlash against corruption (which there would be a lot of them due to butterflies), Rhode Island manages to implement several steps, one by one, to make things more "clean" - i.e. turning the state's Electoral College designation into a
de facto electoral commission and an independent electoral boundary commission is set up, the invention of the
constructive vote of no confidence and the secret ballot, total abolition of the counties, earlier universal suffrage, and other steps like that.
These steps, and others, would turn Rhode Island into a semi-presidential parliamentary "republic" within the Union, as a mix of both Westminster and (West) German models, and hence fulfill the OP. All whilst essentially retaining FPTP at first (indeed, in the case of Rhode Island, if the electoral boundaries are drawn right, then FPTP would result in
de facto PR

). Maybe someone could try going for
AV+ (which is essentially
mixed-member majoritarian, or MMM) later on, but that's a different story.