The primary color I was given to work with is dark blue that is almost black. I need to use this as part of the branding, but what about when it comes to choosing a color for text? I try using black for text but it's so close to the brand color that it looks like using two very similar colors on an interface for no reason. Should I instead use the brand color as a text color?
You have chosen the right page to ask this question, not because of the answers you can get, but because the solution is in front of you. If you go to the questions page of this site, you will see that one of the corporate colors is an almost off-black dark blue, and the title has the same color:
In fact, it is quite rare to find current web designs where the text is black, the contrast is usually softened with an off-black color or dark gray.
In the case that you propose, everything depends on the surface that both elements occupy. If the page is a book with an exaggerated amount of text, and the corporate color occupies a large space, perhaps the text color to choose is another so that the main color stands out (look at the text of the description of the questions on this same page).
Always depending on the design, my decision would be headings (and subheadings) in the corporate color and normal text in a tone with a higher tonal value (lighter than the corporate color or gray).
Thenounproject is an example of a web page with text of the same color as the corporate one (off-black) different from black.
Black (or a very dark color) for text on light colored backgrounds provides the best contrast.
The most contrast between text (foreground) and background provides optimum readability.
Other text colors may be used for bringing attention to bits of text: headers, emphasis, however foreground/background contrast is still paramount and since the dark blue looks similar to black it doesn't provide enough distinction for an attention-getter.
However, using other appropriate, non-blue colors for text emphasis might create more visual separation between the dark blue brand and the black text, and help distinguish the brand.