Definitely check the fan belt for slippage first, good one Bob.
Instead of looking for a spark, connect a test light between the battery and the disconnected cable. If it lights with everything in the car off, then you have a short or ground somewhere in the system. The best way to do this is with an ammeter so you can see exactly how much current is being drawn. Then you remove fuses one at a time to isolate the bad circuit.
If there is no current being drawn in the ignition off position, then you look elsewhere. Are all your connections clean and tight? Don't assume, clean them with an inexpensive battery brush, and make sure all grounds are metal to metal, greased(dielectric grease), and use star washers. Sometimes adding a ground cable from the engine/trans to the frame can help with these problems.
Also, even though your alternator tests OK, what voltage is it putting out when idling and say 2500 rpm? If it is less than 13.8v, that may be the problem. Also, it really shouldn't be higher than 14.5v. Even though the battery is new, test the battery voltage with the car off after you've just driven it, should be slightly above 12V. Test it the next day, and then the next day. It really shouldn't drop any at all, maybe a tenth(.1) of a volt or two. Also, check the voltage while cranking, it shouldn't be below 10 volts. It is entirely possible you have a "new" battery that is marginal. If the same symptoms were present before you swapped the battery, then the problem is likely in the car.
For the winter, I remove my battery and put it on a float charger, or battery tender, indoors on a wooden workbench. The charger is a 1 or 2 amp selectable model, available at Lowe's for $20.
Please report back, I hope we can help you solve your problem.