How to draw ~1A from Li batteries without damaging/over-stressing them?
Batteries have different capacities and the one with the lesser capacity will define how much capacity you get from a series connection of multiple cells.

As you can see in the datasheet you linked, the cells will hold their voltage pretty constant until they drop off a cliff at the end. So I don't think you have a strange behaviour with one cell at 1.47 V and the other at 0.2 V. Both are actually discharged. Here is a relevant document explaining that cells which have an open circuit voltage of less than 1.7 V are to be considered discharged when they had the chance to rest (and recover their open circuit voltage)

But there is a second effect happening which might be the troublesome part for you. As the cell gets discharged, the internal resistance rises. So if one cell is drained the internal resistance might be too high for you to draw all the current you need.

Third thing is that your current requirement will rise during discharge. While it is 1 A at 3 V at the beginning it will be 2 A at 1.5 V when the batteries are almost dead which is overstressing them.

The effect you see in 2. is the battery relaxation effect. I think that the batteries are running dry and you are in the region where the internal resistance is rising. In that state you can extract energy of the battery only with a smaller current.

Depending on your SMPS design, it might draw peaks which are a lot higher than your average current, which will stress the batteries even more. You could try to reduce the problem with a really low ESR capacitor bank which provides the peak currents while the batteries then provide the average current.