There is a limited adjustment path to permanent resident via NACARA, which is limited only to those who entered before 1995 as asylum seekers.
Adjusting from TPS after accruing 180 days or more of illegal presence is the same as adjusting status from no lawful status.
If being sponsored as an immediate relative of a US citizen (spouse, parent, or child under 21) then all that is needed is the original entry to US to have been lawful, ie. you arrived on a legal visa, visa waiver, or parole and were inspected by an immigration official.
If you entered illegally then there's two main avenues:
1) I-601A waiver: your US citizen immediate relative needs to file this form with every and all evidence that your absence from US would cause extreme hardship to him/her. This can be very difficult and often requires an experienced lawyer. If approved the previous illegal presence is pardoned and you have to leave US and undergo consular processing abroad (ie. leave to pick up your visa back to US).
2) Advance Parole: TPS beneficiaries can if they demonstrate a valid work, school, or humanitarian reason apply for advance parole. Advance Parole is a permission to leave US temporarily and return without losing your status. If a TPS holder gets Advance Parole, leaves US, and subsequently re-enters using it, he/she has been legally inspected and paroled into US, therefore he/she can adjust status without need for waivers. Ie. He/she will be treated the same as a visa overstayer.
Any other sponsorship methods are unavailable if you had 180 days or more of illegal presence. Only way to adjust status by family (other than immediate relative category) or work would be if you're covered under 245(i). Ie. you, or someone you were a dependent of at the time, had an immigrant petition filed on your/their behalf on or before April 30th 2001, and the primary beneficiary can demonstrate that he/she was present in US on December 21st 2000. 245(i) coverage allows you to adjust status via any avenue - any family category, any work category, even green card lottery, even if you lived in US illegally for 180 days or more or entered illegally (this applies to above as well, if you entered illegally, but are covered under 245(i) you can be sponsored by Immediate relative US citizen without need for waivers or Advance Parole).
If you have 179 days or less of living in US illegally then all sponsorship methods are open to you, just in all cases except being sponsored by Immediate Relative US citizen you'll have to leave US and undergo consular processing abroad.