If you've performed the isolation test to prove the problem is only the tub/shower valve... And problem not occurring at bathroom sink at same time...
You may have a hot limit stop which you may adjust in order to permit the handle to turn into the hot some more.
If that doesn't help, then yes, replace the cartridge. Most brands call the part a cartridge, but there are exceptions such as Symmons who call it a spindle.
If you decide to replace the cartridge, before you take the valve apart, try to identify the valve manufacturer and specific model number.
The desktop version of delta or Kohler websites have a nice way to assist you identify your model number.
So when I need parts, sometimes I can't find the one or two pieces in stock today. In that case I can usually purchase a complete rough valve with cartridge and they're not expensive and are usually in stock at my wholesalers. More parts before disassembly is better.
Delta valve may have about 6 different cartridge designs. If it's beyond 10 years old I'd usually replace the bonnet nut which retains the cartridge in the valve body, after cautiously cutting the old bonnet nut in the same type of method you would use to drill and split a seized nut off of a bolt stud. The reason I prepare for that with a delta valve is that the bonnet nut often seizes onto the valve body when one of two pitfalls may happen: 1.) wrench tool eggs the nut & valve body out of round.
2.) seized nut torques half of a two piece valve body to shear apart on a Delta valve body older than 1999. Those older valves have three 5/16" copper tubes brazed between the brass Body halves, and must be treated cautiously.