I have tried in the past to make precision cuts using a hand saw and always have issues with the blade sticking or hopping out of the cut.
If by sticking you mean that the friction between the blade and the wood is very high, that could be one of a few problems:
The wood you are sawing is closing the kerf and pinching the blade,
Your cut isn't straight,
The set of the saw teeth is not large enough, so the kerf created by the blade isn't wide enough to allow the saw room to pass without undue friction, or
Your saw teeth are too fine for the wood you're sawing.
For the first, you can try to hold the two sides of the cut apart with your off-hand and see if the saw goes easier. Or, you can stick a little wedge at the top of the cut to keep the kerf open.
For the second, cut straighter! (Note: the cut can wander if the set of your teeth is uneven, regardless if you're sawing correctly)
For the third, ideally you'd use a saw set (pictured below) to increase the set of the saw's teeth.
However, if you bought the saw new at a big box store, there's a high likelihood that it has induction-hardened teeth. These don't bend very well, and will likely snap off if you try it. I can't find a good image of these these of teeth, but they have a slight bluing from the temperature change in the metal. In this case, you might be out of luck.
Finally, for the fourth, you might need to use a saw with coarser or finer teeth. Generally, if the blade is sticking in the kerf, you need a coarser saw. If it's hopping out, you might need a finer blade. Greener wood needs coarser teeth than dried wood.
Some other things to consider:
A dull blade will want to jump out of the cut.
Rip teeth blades sometimes have trouble staying in the start of a cut just due to their tooth geometry. In these cases, I like to start the cut with a crosscut saw to get the kerf established and finish with the rip saw. The link below the image has probably more information that you ever need to know on saw tooth geometry.
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