Unfortunately they currently have contradictory alerts posted for the Red Track:
https://www.nationalparks.nsw.gov.au/visit-a-park/parks/bungonia-national-park/local-alertsClosed areas: Walking tracks closed due to high water levels
Applies from Thu 29 Sep 2022, 10.51am. Last reviewed: Fri 07 Oct 2022, 7.28am.
The following walking tracks are closed until further notice due to heavy rainfall and rapidly increasing river heights:
Red walking track, including Bungonia Slot Canyon
Yellow walking track
Safety alerts: Bungonia Red walking track - Occasional high water
Applies from Fri 03 Jun 2022, 2.33pm. Last reviewed: Fri 14 Oct 2022, 12.56pm.
Whilst there is continuing rainfall periodically in Bungonia National Park, the Bungonia Creek water level may at times be higher than normal.
If you're planning to undertake the Red walking track, please assess water levels before continuing through the Bungonia Slot. If in doubt, turn back. Do not enter flood waters. Report conditions via the park register book on-site.
That said, I have walked down to the Shoalhaven and back on the White Track in the past and it doesn't appear to be closed currently.
I've also walked upstream along the Shoalhaven and returned via the Bee Box track/spur.
But that route needs some navigation and you have to wade in places, so would not be wise if water levels are up.
As WFBW suggested call them to get (hopefully) the most accurate information.