I'd like to assert any claims possible even if I sign the severance package.
No, if you sign the severance package, you will likely waive all claims depending on the language of the Release.
State, Local, and Municipal Law Attorney
Assuming you are an at will employee and there is otherwise no contractual obligation regarding paying you severance, then no you won't be able to assert claims after you sign the waiver. The employer is buying peace and certainty in exchange for your waiver. You can't have it both ways. Consult with an employment law atorney to review the severance package and waIver agreement before you sign anything...
Personal Injury Lawyer
Best thing to do is have a local employer law attorney review your severance package before you sign
This is not intended to be legal advise or as legal representation. I am a California personal injury attorney . Be aware that every state has its own statute of limitations; and statutes & case laws that govern the handling of these matters.
You are not legally allowed to waive your right to make a worker's compensation claim. Thus, if you had a valid worker's compensation claim you could assert it. If the release is properly written, you would not be able to assert other claims.
Employment / Labor Attorney
If you believe you have any valid legal claim, call an employment law attorney to discuss. Most severance agreements will have a clause for a waiver of all claims whether known or unknown. However, you can't waive a right file for unemployment compesation or workers' compensation and age discrimination claims require a set amount of days to let you consider whether you want to sign and then a set amount of days in which you can back out even after you signed. If you have any valid claim, your attorney may be able to use it as leverage to get more money in the severance package.
If you have valid claims, don't sign the severance package.
Consult with an employment attorney who can review the severance agreement for you.