I tend to drop exposures about 1/2 to 2/3 of a stop in tungsten light - and a lot more with some of the new flourescent lights. Haven't checked it with LED lights yet.
Most problems occur when you are shooting indoors in a room. You have bright lights and deep shadows. To keep the light balanced, you usually have to expose for shadows and let the lights fall were they may. Otherwise you have a solid black with 'spots" of bright lights.
The modern films are better, but not infallible. When in doubt - overexpose and be prepared to burn in light sources.
Some developer will work better with this kind of light - flatten the contrast. Stand and semi-stand developing helps, low contrast developers like D23 also will give a less contrasty negative.
Tri X works, but the range is limited - particularly if you are going to 'wet" print. Film has more of a latitude than paper. Scanning and contrast modification in L/R or P/S is easier.
Kodaks XX does indeed give a 200 iso for tungsten and 250 in daylight. Of course, we are looking at staged scenes with big Klieg lights that can be turned up or down to lift shadows and balance the brights.
Watch some of the old classic movies in bl/w and see how the light is balanced - Orson Wells was a master at this "The Third Man" is a great movie in itself - but also a lecture how to light.
If you can, switch films when shooting tungsten and develop it specifically for the situation - either by changing developer or processing.
Also remember that light is not constant - it actually "flickers" usually at 1/60 sec intervals. If you are shooting at speeds around 1/60 you can hit a low glow and shots look underexposed - but another shot, a split second later looks OK.
As with everything, test and note down variations.