Those stick fertilizers can make plants ill.

would agree, concerning for fertilizer burn. Other possibility is not watering enough when time to water to flush out dissolvable salts in tap water so that they build up In the soil.

could we see the whole plant?

@callirhoe123 Here's a photo of the whole plant:

@Tiffany, purpleinopp Z8b Opp, AL It's possible adding the stick fertilizer aggravated the situation - but the plant started dropping leaves with the same symptoms before I added the fertilizer. I've also read about stick fertilizer causing damage, so I broke up a single stick into multiple sections and scattered it around the pot. I don't think there would be enough fertilizer to cause such significant damage, but it's a possibility!

@bob_15 When I water, I water enough such that the soil is moist most of the way through (verifying with a wooden stick I have, see the new photo), and there is some water in the drainage dish that evaporates away after about day or so. Should I change how I water the plant?

I don't know what is causing the browning but, from the looks of it, it is not getting near enough light, hence the straggly stems. If it were mine, I would cut it well back and put it in a brighter spot.

I think it may be overwatered and then the fertilizer aggravated the situation. Salt buld-up is also possible. Slide the root ball out of the pot and check if the roots are white and stiff or brown and mushy. If the roots are healthy slide the root ball back in. Take out the sticks if you can find them. Allow the soil to dry off. then take it to a sink or bath tub and water enough to flush out salt build up-let the water run through a few times. Clean the saucer as well. If all fails, take cutting of healthy portions of the vine, put them in a vase of water, allow roots to grow sufficiently long and pot up in fresh media. Good luck.