My DS was on it for years too OP, he had very similar symptoms to your DD. He also took pico-sulphate at night, and movicol in the morning.
As a PP said the poo you are describing does sound like overflow.
It was so difficult to solve the issue with DS, took seeing two different consultants to get the problem fixed, the first one was awful quite honestly, very off hand manner, just said use suppositories, no way could I do that, it upset my son etc, so we struggled on for a few years.
Saw another second paediatric gastroenterologist was absolutely amazing, got to grips with the problem straight away, give by pressing his tummy, DS had allergy tests just to check that wasn’t the cause. Prescribed meds, mentioned above, that my son was on for quite some years.
THIS IS the important bit….The consultants rule of thumb was, one year of medication for every year of constipation.
my DS is an adult now, never has any issues whatsoever, goes to the loo fine, no meds, no marks in underwear.
so please don’t worry it can and will be sorted out.
if you are on the south coast and want the name of the consultant we saw just PM me, we saw him privately, he was wonderful and changed our lives, I felt desperate by the time we saw him.
good luck 