You may not need a lawyer.... yet. You should have a HUD contract both you and an official with the public housing agency signed. Find it, read it. It will spell out what you can and can't do. But for the most part you can take actions in the same manner you would with a self pay/non-section 8 tenant (and always within the bounds of the law).
If this was a self pay tenant and they caused damage how would you handle it? Would you throw your hands up in the air and say oh well or would you go after them for payment? I'm not saying you will get it, but the question is still valid.
If they say no to a rent increase would you say, well okay that's fine, or would you say something like Regrettably I won't be able to continue to rent to you at the current amount and you'll need to vacate at the end of the lease?
Really anything that you would hold a self pay tenant to applies to section 8 tenants with very few exceptions. Again read your HUD contract, and the lease you have between you and the tenant for some guidance.
The tenant's case worker instructed the tenant that they don't have to start looking for a new place to live because you, the landlord, have not given them notice to vacate. Do that, making sure the amount of notice given at minimum complies with local laws, what is written in your lease, and what is in the HUD contract, if anything to that effect.
For the most part a section 8 tenant won't risk losing their voucher and aside from the damage they caused, an eviction will certainly create a risk. Knowing this should give you confidence, and you can use this to your advantage as leverage.
They may be reluctant to move but the odds are more likely than not that they will move without you going thru a formal eviction process. I would also notify the housing authority of the issues. Or at least use the threat of notifying as leverage to get them to cooperate and move out.