Hello again, Lane,
There is not a whole lot to see in either Dundee or Aberdeen. Dundee has the new V&A exhibition, the RRS Discovery, the frigate Unicorn, Verdant Works (a museum devoted to the jute industry), and a couple of very good art galleries. Most of those are in the central area, and you could probably knock them out in a day.
Most of the sights in Aberdeen are outside the city centre. Central sights of interest include Provost Ross's House, which forms part of the Maritime Museum, Marischal College, Footdee ("Fittie"), and a little bit farther from the centre, Duthie Park, with its Winter Gardens, and King's College Aberdeen. The city centre is a fairly depressed area of closed shops and "Poundstretcher" type retailers. This despite the efforts of the Council to bring new business to the city. Union Terrace Gardens has still not reopened.
There are several castles outside the city, most of which are worth visiting. However, you'd need to take buses to get to most of them, or sign up for a castle tour, of which there are many.
I'm not trying to discourage you from your second option. I have always enjoyed visiting both cities, but without a car to get you out of town conveniently, you could pretty much see all of the main central sites of interest in one full day in each city. As Aberdeen and Dundee are on your way from Inverness to Edinburgh, if you take the coastal route, you may want to spend a day in Stonehaven (for Dunnottar Castle), or Arbroath (for the Abbey, the coastal walk, and smokies). That is, if your ticketing permits. If you're running short of tickets, you could pay for an add-on to either town.
Let us know how your plans come along, and what your final itinerary will be. I'm sure that whatever your final itinerary is, you'll end up deviating from it! And always come back here for advice. Most of us have been to Scotland many times, and are always eager to share our itineraries.
Very best wishes for your travels!
Mike (Auchterless)