How do I determine if the signer is authorized by the client to sign legally binding contracts?
There is no concrete, universal approach for that because the details of how each company is organized can (and does) vary. From a legal standpoint, you would need to focus "only" on the issues of signer's apparent authority and/or your reasonable reliance thereon, since these would determine whether your claim (and under what theories: breach of contract, fraud, unjust enrichment, and so forth) is against the company or the signer.
If you reasonably relied on the signer's statements or acts but are unable to prove the element of apparent authority, the contract might not be binding to the company but you would still have a claim of fraud. That is because the prima facie case of fraud involves (1) a plaintiff's reasonable reliance (2) on the defendant's intentionally false representations that (3) led to the plaintiff's loss. Your claim of fraud would be against the signer, not the company. Depending on the jurisdiction treble damages might be awarded.
As for apparent authority, it "arises where the acts and appearances lead a third person reasonably to believe that an agency relationship exists. However, apparent authority must be traceable to the principal and cannot be established only by the acts and conduct of the agent", Alar v. Mercy Mem. Hosp., 208 Mich.App. 518, 528 (1995) (citations omitted). The notion of traceability to the principal is also mentioned in the Restatement (Third) of Agency at § 2.03.
Being able to prove [in court] the signer's apparent authority would, in theory, lead to a ruling against the company because "[a] principal is bound by contracts made by its agent, provided the agent is acting within the scope of his actual or apparent authority [and] an agent acting within the general scope of his apparent authority, though exceeding his authority, binds his principal". Clark v. Tennessee Farmers Mutual Ins. Company, (April 17, 2020), citations omitted, emphasis added).
Thus, the scenarios where the signer exceeded the limit in the amount he may authorize, or where signer's authorization has been withdrawn would still bind the company unless it can prove that you knew --or reasonably should have known-- of the constraints of the signer's authorization or what the status thereof was at the time of the formation of the contract.
Even if the company persuasively refutes your argument of signer's apparent authority, the company's or employees' use of your services would still lead to a claim of unjust enrichment insofar as they benefited from your services for no apparent reason other than your contract (for instance, it was neither a gift nor an entitlement of theirs to a benefit from you).
Lastly, the amount owed to you per a contract might be such that you could only pursue your claim(s) in Small Claims court. Some (or many) jurisdictions parties are not allowed to be represented by an attorney. This means that you might not have to worry about having to feed a lawyer in the first place.