The flower is dieing and needs removing. If you don't like the sticky aspect, leave it for a couple of days and the dead flower will be dry. Removing the dead flowers - or 'deadheading' - prolongs the flowering period. Once a plant sets seed it thinks its job is done and stops producing flowers, so removing the dead flowers along with the seed keeps it going.

I disagree Petunias have naturally sticky leaves/stems, so don't panic and the limpness sounds like lack of water? So try giving them a good soaking twice a day and with Petunia surfinia a little light shade may be of benefit - its been very hot/dry, and as you dont mention a disease such as mold you have nothing to lose?
Good luck.

Agree with Drc - its the nature of the beast, petunia leaves, stems and flowers do become sticky, usually by mid to end of June. And the limpness will be a lack of water.

I'm so glad you both said THAT! I have baskets and pots full of them...and yes!...they are decidedly sticky sometimes.