My first ever celery too - also self blanching - planted quite densely. Went soft within two days of picking but I had forgotten to put it in the fridge, though perhaps in water would be better.
They started off just producing stacks of leaves but then the stalks seemd to get going (above ground) - so perhaps you should give them a bit longer?
I thought they'd be useful since I understood they could be left in the ground for some months. Just read the self blanching varieties aren't frost hardy but I'm in central London so fingers crossed.
I like putting it in the bottom of the roasting pan with onions and carrots to give extra oomph to gravy. Makes great soup too.