I'm relatively new to gardening, and tomatoes have never been my strong suit.
One of my plants is yellowing from the bottom up, about 1/4 of the plant. The edges of the leaves are browning with some small dark dots here and there. It appears that the veins are the last thing to yellow. Also once the leaves yellow the stems yellow and most of the time drop off. However, fruit has set and is producing ok it seems.

This particular plant is in a large container, probably about 10+ gallons.
It's been humid with flucating nighttime temps. We've had a few big rainfalls so far. I may be overwatering. It has had Epsom salt about 2 weeks ago and a balanced organic fertilizer 2 weeks ago as well.

Can you post a photo of the whole plant? Why did you add the Epsom salts? If you used commercial potting soil and a well balanced fertilizer, Epsom salts are not necessary.

You may have a fungal infection.
When there is excess water, either from rain, humidity or sometimes overwatering...a fungal infection can take hold. There are products you can spray on the foliage to help eliminate the fungal infection if that is what it is.

I agree with Daisy; a photo of the entire plant would be helpful.

Don't mind my busted up totes. It was a last minute "im broke" idea after an out of the blue move. I used Epsom salt because I had thought it was a magnesium deficiency at first. Well, I suppose I had hoped it was so it would be a simple fix.

Daisy, you are partially correct. Magnesium (Epsom Salts) are NOT necessary IF the plant is in the ground, since the ground usually provides the tiny amount of magnesium required.
But. . .
Not so if the plant is in a container, as in this case. Since the magnesium atom is at the center of the molecule for photosynthesis, it is a recommended practice for growing container tomato plants to apply a foliar spray of unperfumed Epsom Salt (MgSO4) at a concentration of 1 tablespoon per gallon water, especially on the lighter green new growth. Wet the newer growth thoroughly, but not so much that the leaves are dripping: do this about every bweek or so, as needed (indicated by the tell-tale lighter green top/new growth).

Stephanne, comparing your photos with the following 2 links lead me toward concluding your poor plant is starving!! It appears to be suffering from a nitrogen deficiency; the cure is simple. . . . FEED IT! The 3 numbers on fertilizer analyses are Nitrogen-Phosphorous-Potassium. Fertilize more often or change the fertilizer you're using to one with a higher N content.
http://www.haifa-group.com/kno...
http://customers.hbci.com/~wen...

North Carolina has pretty active tomato help , especially thru the state universities, county Agriculture Extension agents and Research Stations associated with the universities.
https://projects.ncsu.edu/cals...

Here's a link to common tomato diseases and cures (YouTube).
https://www.youtube.com/watch?...

Another YouTube video about foliar spraying with baking soda to combat fungal diseases
https://www.youtube.com/watch?...

I grow tomatoes in containers. I'm always learning new things, as I'm still no expert..but I'm willing to help if I can. Feel free to shoot me a Tree mail.

Here's a great article on the use of Epsom salts on plants, both in the ground and in containers.
https://garden.org/learn/artic...

My opinion is it's not a nutrient deficiency that's causing your leaves to yellow and wither from the bottom up. If it was, the whole plant would be affected. A plant that big in a container certainly does need a constant supply of fertilizer though. The bigger it gets, the more fert you need, and weekly rather than every 2 weeks as the weather warms might be a good idea.

Much more likely cause IMO is either a Fusarium wilt or blight disease. They're tough to combat, but I've had a bit of success dousing the whole root system with a very mild peroxide douse. Drugstore peroxide diluted 1oz. to 32oz. of water and pour over the whole root system (wet all the soil in the container). It's hit or miss, but might just catch the plant before the whole system is affected. The douse is mild enough that it certainly won't hurt, anyway.

A disease???? Hmmmmm.

I still disagree, but let's let Stephanee diagnose her tomato problem herself. To help (in addition), here's 2 keys :
University of Florida Tomato Disease Identification
http://erec.ifas.ufl.edu/tomat...
Cornell University Vegetable (tomato) MD Online
http://vegetablemdonline.ppath...