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Special Relativity doesn’t actually say that nothing can move faster than light. It says energy (which includes matter via the famous E=mc2 equation) cannot move faster than light. Quantum entanglement doesn’t involve any actual motion of matter or energy at faster than light speed. Observer A can observe the state of his entangled particle. We know that the state of observer B’s particle is determined by this measurement, but this doesn’t involve the physical motion of any matter or energy. It is analogous to an experiment where I send one glove from a pair to two different people. When the first person looks at his glove, we immediately know which glove the second person has, but nothing has moved faster than light.

We also cannot use such a method for FTL communication. The glove analogy also is illustrative here. The first person cannot use a series of “left” and “right” gloves to encode a message since he has no control over which glove he gets. Similarly when an observer measures an entangled particle, he measures either of two states (usually a spin up or spin down state), but the value that he measures is random. Observer A cannot send a meaningful message since all he can do is generate a random series of up and down states.

Observer B will have his measurements determined immediately by this, but unless he knows he is observing an entangled particle, it will appear to be a completely random set of observations indistinguishable from what he would observe with a non-entangled particle. Only by actually communicating with A (using normal light speed or slower methods) would he see that his particle is entangled with A’s

You do almost hit on the interesting an most problematic part of modern physics though. We know that within its domain quantum mechanics always gives the right answer. We also know that within its domain General Relativity always gives the right answer. The really interesting and problematic thing is that we also know that NEITHER of these is a complete theory. They both give right answers, but they are completely incompatible with each other.

We know that quantum mechanics is completely compatible with SPECIAL relativity. That is it works at high energies so long as gravity can be safely ignored, or alternatively QM works on flat spacetime. QM cannot be used when gravity is non-negligible, that is when spacetime curvature is important.

Remembering that the domain of overlap for these two is very small distances and very large gravitational fields, it is easy to see why we did not realize that this was an issue at all for most of the history of physics. The systems we use QM for are ones like molecules, atoms, nuclei, and subatomic particles, all of which have very small masses, and thus negligible gravity. The systems we use GR for by contrast tend to be very large ones, like stars, galaxies and even the whole universe, ones for which quantum effects are negligible.

There really are only two cases where there is overlap. The Big Bang cosmology posits a very small dense initial state, small enough to require QM but energetic enough to make spacetime highly curved, requiring GR. The other is black holes; we can do well enough outside the event horizon with just GR, but inside the event horizon, we are dealing with the possibility of all the mass occupying a single point - the singularity, requiring QM.

The two appear to be non-reconcilable; trying to do quantum calculations on curved spaces leads to nonsensical answers. Likely there is an overarching theory that would incorporate both of these successful theories. String theory was the first attempt at such a theory and showed promise initially. The calculations gave good answers and appeared to account for both domains, but experiments have refused to cooperate. Most string theories were based on ideas known as supersymmetry, in which the known particles in the universe had massive “supersymmetric partners”. The masses of these partners were calculated, and we built particle accelerators that could reach energies sufficient to generate these partners, but they failed to show up. It’s possible that there still are supersymmetric partners at higher mass, but the presence of such particles would no longer allow reconciliation of QM and GR - back to the drawing board.