You may find a little rearranging of your schedule is in order. If post-exercise damp hair is a problem, you might have to change your workout times. It's not always the ideal solution, but neither is going to work with wet hair. You'll need to ask yourself if you can either (a) exercise earlier, thus giving yourself more time afterward before heading to the job or (b) exercising after work. Even so, in some cases it can take thick hair hours to fully dry without help.
Your next step is to consider low heat. Yes, frequent and/or high heat is not good for black hair, but the occasional use of low heat -- preferably in the form of a diffuser dryer, bonnet or hood dryer -- can speed up drying time without wreaking havoc on your locks. To dry most effectively, use your fingers to circulate the air more quickly. You should still stop short of completely drying the hair; you may choose to turn off the heat once the hair is no longer dripping wet or you may want to dry about 80 to 90%.
During the winter, it's more important to avoid walking around with wet hair than it is during warmer weather. This is more vital for people who live in extreme climates, so even if you steer clear of any kind of heat the rest of the year, consider it when it's cold out.
Lastly, wear styles that dry more quickly. Hair that's confined will hold in moisture, so buns, chignons, braids and twists take longer to dry than hair that's worn loose. This may take some planning when deciding on your style for the day or week, but as anyone who has a hair care routine knows, planning is part of the regimen.