If it's been a week or so with little/no water in the cooling system I'd imagine the engine has been well and truly 'cooked' by now.

If the oil looks light brown/beige and the consistency of mayonnaise in the filler cap its an indication of short journeys (normally), where the engine hasn't got hot enough and condensation has built up emulsifying the oil mist with the condensate in the top of the engine.

In your instance though, it seems to be a bit more serious than just a bit of Mayo in the rocker cover ! If the engine sounds rough and it is indeed steam coming from the oil filler cap, it does sound like a possible head gasket and/or warped cylinder head itself ?

Get a garage to carry out a head/block leak test with the blue indicator fluid test equipment and I'd also suggest a compression and cylinder leakage test too.

If it fails either test it's a head off job to inspect for further damage I'm afraid. Hopefully, it's just the gasket, but worst case scenario is a warped/cracked head !