Is there any textbook which teaches abstract algebra and programming concurrently, focusing on perhaps the applications of abstract algebra to various computer-science related fields?
Not exactly algebra and programming, but Meyer, Lehman, Leighton "Mathematics for Computer Science" (link is to lecture notes from which the book was forked) covers much of the mathematics used. Little algebra, very little programming as such. For a deeper look into algorithms, check out Erickson "Algorithms" (again, lecture notes that gave a book). Heavy on algorithms, lighter on math.
For programming as such, look for a good text for the language that most interests you. You might also be interested in the books by Allen B. Downey, "Think ...", he wrote versions for Java, Python 2 and 3. They are available for free as PDFs too. They cover more of the "why" of doing something as of the nuts and bolts. Other good texts, not language bound as the previous ones, are Bentley's "Programming Pearls" (2nd edition), and Kernighan and Pike, "The Practice of Programming".