Ok I can report back now after riding with the saddle back for some time -- interestingly that seems to help prevent the forward slipping, and lets me sit with the seatbones on the saddle nicely planted. My hypothesis is that my position was too far forward in the first place, so the centre of gravity was not properly over the saddle.
Interestingly I also find it very easy to spin the pedals in the more back position, which seems to be contrary to popular belief that setback is better for mashing. Anyway, probably just another indication that my initial position was not optimal.
Haven't done any climbing yet, though. The mountain roads are not in condition yet.
The guidance for finding my new position is Steve Hogg's "Point of balance" -- http://www.stevehoggbikefitting.com/blo ... oad-bikes/
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Next step: "How to ride in the drops like the pros", my goal is being able to comfortably do "all out" over 20km in the drops within a few months, and over 40km in fall, for taking part in TTs with a road bike classification. Might open another thread for that.