Bitter Basil

My basil I started from seed is finally matured. I only have one variety growing (I couldn't get the other varieties to germinate). I started them in store bought potting soil, and all are outside in containers now with plenty of sun and rain.
I tasted the basil this morning and it had a very bitter flavor, not the flavor I expected to get. Also the plants have dark brown/black stems, the bottom leaves are yellow, but the tops are beautiful.
Are they getting too much water? I live in Kansas (near the Missouri border) and it has been raining 5 of 7 days for more than a month
Do they have the fusarium wilt? If so what are my options, and how do I prevent it in the future?

They might have an early case of the wilt, and that can affect the flavour, but it is also likely that the plants are not getting enough sun to develop the proper basil flavour. The excessive moisture may also be creating the conditions for other fungi to flourish on the leaves, such as downy mildew.
In the future, I would suggest that you order fusarium resistant varieties of basil, because if your plants do have it it will from now on be in the soil and would probably kill the newest plants much earlier. It might also account for your lack of success with other, possibly more susceptible varieties of basil.
I would suggest that you put a bit of a rain roof of plastic over the seed rows – copy English gardeners who have to cope with this almost every year. Make sure you have good drainage by augmenting your soil if necessary to be a sandy loam.

Should I fertilize differently? I am using an all purpose plant food from Schultz (20-30-20).

Enriching the soil with compost and well-rotted manure may be better. It results in less rapid growth, but the growth will be harder, more disease resistant and more flavourful.

Lastly, can basil's flavor be affected by plants planted nearby?

In theory yes since all plants secrete chemicals into the soil to cut down the competition fom neighbours. Some secretions are more deadly to selected plants than others. Soil composition has a definite effect on flavour and you might experiment with different soil mixes to help with your poor flavour problem.