Doesn't entanglement mean that entangled particles just cannot have same state of the entangled quantum property at the same time, but they can still achieve all states, essentially preserving their degrees of freedom
A state is entangled when it isn’t a product state.
Two spin (1/2) particles in a singlet state have the kind of “they have opposite states” thing going on that you describe, and is a specific way that two particles can be entangled.
A state like (01 + 10) is not separable, so by definition it's an entangled state. "Separability" is a straightforward algebraic fact that follows from the definition of a vector and the tensor product. You can see what this means in the following Google answer
https://share.google/aimode/13jNpR7bmpPMo1pn3
(01 + 10) means that if I measure the first particle and get 0, then the second particle is now in the state 1. If I measure the first particle and get 1, then the second particle is now in the state 0.
The state sqrt(1/2) ( |00> + |11> ) is also possible, and is also an entangled state, but doesn’t have the two particles in opposite states.
By contrast, the state (1/2) (|00> - |01> + |10> - |11>) is (while a valid state) not an entangled state, because it is equal to (1/2) (|0> + |1>) (|0> - |1>) .