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Petunias have naturally sticky leaves and stems, so don't panic, this is normal. As for fertilizing rosemary plants, I would actually recommend not fertilizing a rosemary. Rosemary and many drought tolerant herbs actually grow with a stronger and better flavor if not fertilized.

Since your zone is 6a-6b, planting annuals should be safe after May 15. Mandevilla is a tender annual. This article should help with growing the plant: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/mandevilla-plant.htm

Are they getting plenty of light? Without enough light, they will be more leggy and act this way. These are, in fact, a trailing type petunia and should begin cascading over the rim of your flower box. You can try pinching the growing tips to make it more bushy, which should help it quit growing up.

You're using too much fertilizer; plants in pots need to be treated very differently from plants in the ground.. Fertilizer should be used at 1/2 or 1/4 strength, and no more than once a month. You can run water through the pot (5 times the volume of soil) to wash out excess salt, which comes from a build up of unused fertilizer, and don't fertilize again for 2 or 3 months. Also it would probably be good to increase the light if you can. Potted petunias need several hours of sun a day, if ;possible.

Here in Florida, petunias are only a spring flower, because they can't take the heat. If you've had a lot of heat there, this could be your problem. Also you could be watering too much, if the soil is not extremely well drained (read no "moisture retentive" products in it.) Here's an article on petunia problems for your consideration: https://www.gardeningknowhow.com/ornamental/flowers/petunia/problems-with-petunias.htm

The most likely culprit is budworms. This article discusses them: https://www.gardeningknowhow.com/plant-problems/pests/insects/controlling-budworms.htm