Keeping a cheap, DIY transformer safe
The reactance will not be increased significantly by increasing the number of turns on the primary
A common way of making a transformer short-circuit proof is to reduce the coupling between primary and secondary to increase the leakage inductance. This may involve the way the laminations are arranged or even put the primary on a separate limb of the transformer with a shunt path between the primary and secondary.
Increasing the number of turns on the primary will increase the magnetizing inductance and reduce the no-load current.