One way to look at is, if you try to rebuild it and it doesn’t turn out well, all you’re out is the cost of the overhaul kit and float and your time. You could still buy a rebuilt unit and turn your old one in for the core.
Carbs are easy to blame, because it’s often difficult to prove that they’re not the problem. I don’t know what the symptoms are or what diagnosis has been done, but my general advice is to be as certain as you can that other possibilities are not the real culprit.
The secondary barrel is running dry. I’m not quite sure what you mean here. I assume it’s a progressive 2-barrel (one primary venturi, one secondary). On these, you can normally see some gas going through the primary at idle and while revved. But usually you won’t see gas going through the secondary. It’s closed except under load with the primary opened more than halfway. With the vehicle sitting still, it’s not going to have much load, and it can get scary trying to rev it up enough to give the secondary a chance to start working. If the secondary was inoperative for some reason but the primary still worked, I would expect it to run more or less OK as long as acceleration was gentle. I would also expect it to be pretty anemic when you put your foot in it.
The typical problems with this type of carb are dirt/debris build-up, wear of soft parts (e.g. accelerator pump), and float sinkage. The overhaul kit won’t have a float, it would be wise to get one. Carefully study the instruction sheet that comes with the kit to get an idea of what you’ll be doing. Keep in mind that you won’t need to disassemble the carb completely–don’t automatically remove every separate part shown in the drawing.
I’ve had fine results using a couple of cans of spray carb cleaner. One advantage of this is that you can put the thin nozzle into every passage and verify that they aren’t plugged. I follow this by spraying with compressed air (not absolutely necessary but a big plus if available). The soak type cleaners are more effective at getting the parts really clean and shiny, but they’re expensive and somewhat caustic. In most cases, they aren’t needed to do the job adequately.
I won’t be able to cover every base, but here are some tips that come to mind:
Have a large work surface, and lay out the parts you remove in order.
Make any notes or drawings that you feel might be helpful for reassembly.
Note that some adjustments (e.g. float level) need to be made before completing reassembly.
If you remove the jets for cleaning (I do to be sure there isn’t crud built up in the passages), make VERY careful notes about which jet goes where. There should be a number on the jet to help identify it. Do not assume that the primary always get the smaller jet of a pair, sometimes the secondary does, and there is no consistent rule about this. Write it down during disassembly.
Avoid subjecting any rubber or plastic parts to the carb cleaner.
Avoid destroying linkage fasteners (clips, cotter pins, etc.). Sometimes the ones in the kit don’t fit right, and these are itty-bitty things that you aren’t going to find in any store.
If the secondary is opened by a vacuum diaphragm, bench test it to make sure it works. Also test the choke pull-off(s). Most kits don’t have these diaphragms.
A set of drill bits, every 1/32nd up to 1/4", can be very handy for measuring choke unloader, choke pull-off, etc.
Good luck!