What are the options for hanging sheetrock or panels on a ceiling in the basement when electric armored cable and pipes hang below the exposed joists? I know I can use a drop ceiling but it would make my basement feel cramped because I have a minimum amount of headroom. Thanks for any tips.

Add 2x4 furing to undeside of joist to clear electrical:)

You might not get a great solution since 2x4 added as a furring is pretty much now a drop ceiling. Do not notch out the bottom edeg of a joist, it is not like drilling a hole in the middle of the joist and will weaked the board too much.