I have these dark brown spots on my tomato leaves that appeared over the weekend. I'm only seeing them on one branch. The rest of the plant looks fine. The nearest I can tell, it looks like spotted wilt. The leaves are starting to draw up and die. Should I cut this branch to avoid spreading? If not, what's the best way to get rid of it?
Definitely, cut that branch off (clean your pruners with alcohol, before and after). Based on this picture, it looks like you had aphids first, then their honeydew and the terminal leaf looks like leaf spot, a fungus. The spots are going across the veins and there is a slight tan halo where the plant is cutting off nutrition to the areas where fungus is producing spores. The plant does this on purpose to get rid of the fungus.
It's hard to believe that only this branch has been infected. Look carefully at new growth, under leaves and watch the stem. If you get black spots on your stem, the plant is in trouble.
Is this plant outside in the garden? If it gets humid, provide your own wind with an electric fan. If you plan to have more humidity, you might want to spray with Neem, at night to protect bees (some Neem brochures say it is hazardous to bees and others say it is not). Neem is like a little raincoat that is pretty safe...read the directions before you leave the store. Make sure the formulation is safe for edibles. Not all Neem products are equal. Don't water late in the day. Start looking for Powdery Mildew, slight cupping, white dusting...this you will need to get on top of right away. Spraying with Neem or even Serenade (bacillus subtilus, a bacteria that helps prevent fungal infections).
If you start getting into fungal infections you might want to put up a roof to keep plants out of the rain. Still need aeration, so keep the fans under the roofing material. Roof material could be clear plastic, a sort of partial greenhouse.
Do you know what the difference between reproductive branches versus vegetative? What you are holding is vegetative. Meant to gather sunlight and do photosynthesis. If you can see the difference, look for vegetative branches that don't get a lot of sunlight and prune those off to increase aeration. Allowing air to flow through the plant and keep leaves dry.
Please stay in touch, let us know what you find, what you decide to use and how things turn out. Too-wet conditions can really mess up your garden.