Question: If the western chart/tropical shifts the zodiac back a bit do the naksatras also not change in terms of the zodiac sign they are in?
[Actually the sideareal zodiac shifts backwards about 23º from the topical anchors by applying ayanāṁśa.]
The nakṣatras are stars, not signs. We apply ayanāṁśa to the tropical anchors to locate the positions of the fixed stars, nakṣatras. The two, stars and signs, are not identical, the stars shift position gradually against the tropical anchors.
Question: So, for example if I am aries rising and cancer moon in my western (which I think I am) then am i still punarvasu nakṣatra?
No. Let’s say your Moon is at 1º Cancer, tropical. It would be a mistake to consider this as Punarvasu Nakṣatra. It was roughly 2,000 years ago that 1º Cancer corresponded to Punarvasu Nakṣatra. Since then the stars have drifted from the tropical anchor points (solstice, equinox) quite a bit. Today, the tropical position 1º Cancer corresponds to the tail end of Mṛgaśirśa Nakṣatra.
Literally, Punarvasu are the two stars that form the heads of the constealltion known in the modern west as Gemini, the stars are named Castor and Pollux in the West.
The problem is a confusion in where ayanaṁśa should be applied. It should be applied to the stars (nakṣatra), not to the twelve ecliptic divisions based on the equinoxes and solstices (the “signs” / rāśī). This is very confusing for people with contempoary “traditional” education in Indian astrology, where it sounds like a very new and radical idea. However, this is the verdict of Sūrya Siddhānta and similar texts. See: Posts on Tropical / Sidereal Issues
Question: Is punarvasu not ruled by Cancer’s Moon now?
Punarvasu was never “ruled” by any planet. This is a wrong idea contempoary astrologers have come up with over the past few centuries. Punarvasu and all the Nakṣatras are ruled by devas (gods). Punarvasu specifically is ruled by Goddess Aditi.
Question Would this change the naksatra compatibility?
No, Nakṣatra compatibility calculations are done relative to other nakṣatra. Rāśī compatibility calculations are done relative to other rāśī. There is not overlap.