How to set up load sensor in a full bridge with amplifier?
It's likely that on each corner of the scales the cells are half-bridges not quarter bridges. The only 3-wire quarter bridges I've seen are for when trying to cancel long-wire volt drops.

You wire the four half-bridge load cells like this: -



But you have to take care you get the wiring polarity correct or you can end up with weight cancellation and no signal. You just have to experiment a little but that's the basic wiring.

For the first diagram you showed, each half-bridge is excited in parallel with the other half-bridges and two half bridges are paralleled for each input. It does the same as the above but is less sensitive (or put the other way has a larger full-load capacity).

As for amplifying the signal I'd use an instrumentation amp (inamp) like this: -



It's basically 3 op-amps connected as below: -



There are plenty of options and optimization or best choice of inamp relies on knowing the power supply you have available for your project.