I understand that the act of measuring a particle actually changes it. My question is, is that because our method of measuring is crude, and is it theoretically possible to determine the spin without changing it,
There is nothing like (yet) spin measurement at quantum level. It is spin alignment, or spin anti alignment.
Therefore, needless to say, alignment/anti alignment would not be possible without changing it unless the particle already (coincidentlly) happens to be in that state, which you would not be able to know, unless you had already aligned it that way via previous attempt(s).
There is no act of spin measurement, rather it is act of spin alignment/anti-alignment.
is it theoretically possible to determine the spin without changing it,
No, the act of measurement itself perturbs the state of the particle. After you're done measuring the particle's spin you'd never be able to answer what the state of the particle was just before measurement. In quantum mechanics, we don't measure one particle because that doesn't give you any thing. The measurement is done on a collection of the same particles, called an ensemble, which are in the same state.
For example, say there are only 3 possible states that the state wavefunction can collapse into. If you measure the system 1000 times and you get particles in state A for 50 times, the same in state B, and finally 900 times in state C, then the probability of finding the particle in state C is 90 percent. That's how a measurement is done in quantum mechanics.
Also according to QM a particle doesn't have, for example, a precise position just before you measure it, it's the measurement process that restricts it to one particular state, which is determined by the statistical weighting of the wavefunction.