The choice is about more than price
Dorms are often framed as the budget option and private rooms as the comfort option. That is partly true, but it leaves out the underlying structure.
Dorms distribute cost by sharing space, noise, access, and uncertainty. Private rooms concentrate cost in exchange for control over those variables.
What changes for solo travelers
Because you are traveling alone, your room is not just where you sleep. It may also be where you regulate stress, secure belongings, take calls, recover from transit, and regain decision-making capacity.
The useful question
Do not ask only which one is cheaper.
Ask which one fits the role accommodation needs to play on this particular trip.