So, ready to have fun with something weird?
Last year a new toll road opened south of Arlington, build directly in the median of pre-existing TX 360. It was long billed as the "360 tollway" and any logical person might assume that with its construction it would become the main lanes of TX 360. But, they would assume wrong. To wit:
https://www.google.com/maps/@32.6413794,-97.063861,3a,15y,224.85h,91.39t/data=!3m6!1e1!3m4!1sAxs0rQF8Pnya0wMhZeaWAg!2e0!7i16384!8i8192So, it seems pretty clear that the 360 Tollway is meant to be considered a separate and distinct thing from TX 360, which has not moved and remains the designation for the frontage roads which already existed in place before the tollway was built.
This is similar to how TX 121 and TX 161 are the state highway designations for the frontage roads of the Sam Rayburn Tollway and the President George Bush Turnpike, while the tollways themselves have no state highway number and are maintained by NTTA, not TxDOT. But this case is wonky because the toll road's name is simply "360 Tollway"
(there's even a sign that calls it that in text), with the NTTA toll shield containing "360" as shorthand for that (along the same lines as "SRT" and "PGBT" being used in the shield as shorthand for the prior two examples).
In the SRT/121 and PGBT/161 cases, we've mapped the state routes and the tollways separately (the latter in usasf), with the point coordinates offset slightly to avoid auto-detection of a concurrency.
It would seem that a similar treatment may be warranted here.