THE FOLLOWING REPLACES #4, which I am no longer able to edit:
A state function does not define the state of a system. You need to define quantities properly.
The state of a system is defined by the values of the largest number of independent state variables for the system. You are probably at the moment only considering the simplest type of thermodynamic system, called a simple hydrostatic system. The equilibrium state of such a system is fully defined by specifying two of the three variables P, V and T. The variables P, V and T are not independent variables. they are related to each other by an equation, called the equation of state for the system. Three variable related by one equation means that only two can be assigned values independently. The value of the third is determined from these two values by the equation of state. This is expressed by saying that two of the variables are independent, and the third is dependent. So the state of a system is defined by specifying the values of T and V, or of T and P, or of P and V.
A state function f is a quantity that is a function of the variables that define the state of a system. Suppose we choose to define the equilibrium state of the system by the values of T and V. Then a state function f is any function of T and V. Thus any function f = f(T,V) is a state function.
In mathematics, the values of all variables are dimensionless. Their values are pure numbers. We have an added complication in physics: the variables usually are not dimensionless. For example T is measured in Kelvin, and V is measured in cubic metres. So all expressions that we form must be dimensionally consistent as well as mathematically correct.
If f and g are state variables measured in the same units, it makes sense to add or subtract f and g. Their sum is then f(T,V) + g(T,V). This sum is also a function h(T,V) of the state variables T and V, and so is a state function. If their dimensions are different, however, it makes no sense to add or subtract them. However, you could multiply or divide them to get a new state variable with dimensions different from both f and g.
Enthalpy is a state function. Therefore H = H(T,V). Also entropy is a state function. Therefore S = S(T,V). However, they are measured in different units: S is measured in Joules per Kelvin, while H is measured in Joules. So even though they are each state functions, adding them is not dimensionally consistent. It makes no sense to add or subtract them. This means that you cannot form a new state variable by adding them or subtracting them. You could however form a new state variable by multiplying or dividing them.
Since S is measured in Joules per Kelvin, and T is measured in Kelvin, you could form the product TS. The product TS is then a function of T and V and so is a state variable. It is measured in Joules. H is also measured in Joules, so you could form a new state variable by taking TS + H or TS - H.
Note how the above was PROVED. Begin from the definition of the thing in question. Then use the definition of the other quantity to show that it fits the definition of that thing. You cannot prove anything without using definitions in conjunction with known proven results. So make sure that you memorise all definitions. You will not be able to do anything without them. Definitions are key information. Memorise all key information. When doing physics, you have the additional constraint that every expression or equation that you write down must be dimensionally correct. So always also check all the dimensions.
Remember that physics and mathematics are precise subjects. You need to develop precision in your thinking. Vague, wooly, touchy-feely "understanding" is not understanding at all.
You may be wondering if everything in thermodynamics is a state variable. The heat intake into a system is not a state function. Why? Because it depends on the process (path!) followed in going from the initial to the final equilibrium state. Similarly, the work done by the system is not a state function.