Oh boy, lots of question here! Let me see if I can keep from going on for three pages...

So you asked:

How could they evolve?

My first thought is that they started out in a temperate or tropical environment that heavily favors r-selected breeding. Lots of food, great weather, some major predators that like to eat them but the carrying capacity of the environment is never a real limitation on their ability to reproduce like bunnies. Then something Really Bad Happens that destroys some but not all of this environment and makes it inhospitable for everything but the best-adapted life. While it gets most of the proto-critters in that area, it also wipes out their major predators, and perhaps introduces some kind of novel resource (a high-energy food, perhaps) that lets any slightly smarter proto-critter that wanders into there occasionally score a huge jackpot in the survival lottery.

Now let's make the phase transition between the really nasty wasteland and tropical bunny-town abrupt, but not at all impassible for the proto-critters and their predators. Some of them are going to keep wandering into the wasteland, most of them will die, but the extremity of the environment begins working on them. Proto-critters that survive the wasteland start getting progressively smarter, but can still wander back to the more temperate zone to sow their wild oats and thus keep gene flow going between the populations.

I think you could potentially see your S phase crop up among the wasteland-dwellers, in this scenario. They'd still be living in the nastier, more dangerous area in the present day, but they'd go back to the temperate one to breed, since the wasteland will kill off something like 99% of the T phase, versus the more "moderate" 75% that get culled in the temperate zone. This separates the S phase and the T phase out into two separate niches that they can uniquely exploit (T phases reproduce faster, S phases benefit from being less predated upon and having [some resource] out in the wasteland in relative abundance).

The ability of S phases to induce T phase development might show up later.

This is, of course, a just-so story, but a lot of evolutionary origin stories are.

Some other stuff

In no particular order...

1. You might want to introduce some substance that adult S phases produce that induces transformation in T phases. Think of royal jelly in bees--it's fed to all castes of bee initially, but it causes a transformation into a queen in larvae that are continuously fed it through the mechanism of the protein royalactin. Actually, now that I think about it, this gives another evolutionary option for your critters: They used to be eusocial, but the reproductive caste actually evolved sentience (rather than just getting bigger and progressively more useless and reproducing all the time), while the non-reproductive caste got their gonads back. In this situation, S phases may actually exploit their non-sapient T phases to forage and feed them. Everybody wins. This wouldn't be domestication per se as it would be the exploitation of the old eusocial patterns--even though T phases can now breed on their own, they still have psychology that makes them congregate around a related S phase and share food.

2. Ability to recognize near-kin through scent and other markers (perhaps genetically transmissible behaviors like fixed action patterns) isn't a really crazy idea, and it's already especially pronounced in eusocial species (because it lets them avoid having other individuals cuckoo into their nests--though it's not infallible). So it's completely plausible that S phases are very good at smelling out their own children, and it's also plausible that there are some S phases "working on" (i.e., evolving) a reproductive strategy that lets them cheat and get another S phase pair to raise their T phases to sentience.

3. I think your idea of S phases breeding to T phases is actually a pretty good one for continued gene flow. Another option might be that S phases see their T phase grandchildren (great-grandchildren, great-great-great grandchildren), rather than direct offspring, as valid targets to raise into sentience. This works especially well if each S phase has some T phase that naturally congregate around it; they might notice one of their nth-generation descendants is already naturally very clever and decide to uplift it.