Ohmic resistance of the cold bulb doesn't say much. If the bulbs light up, at least for a while, and they keep doing that time after time after turning the projector on, then it's safe to assume the bulbs as such aren't the problem, provided that the correct type is installed.
IDK what kind of safety mechanisms there are present in this projector that might cause it to switch between bulbs and/or shut down. Possibly it has something to do with failure of such a circuit, causing a safety feature to engage inadvertently. Evidently, the projector is equipped with failure detection systems and it seems likely (at least to me) that the problem is related to these.
I'd start by cleaning out the machine to ensure that no dust etc. is affecting dissipation of the heat of the bulbs. Thermal problems can very well explain this kind of behavior and given the age of these machines, buildup of dust and grime seems plausible.
And welcome to Photrio!