The other driver's insurance company has offered the policy limit. There is no way to make it pay anything more than that. The only way to get more from the other driver is to sue the driver, get a judgment against him/her, and then collect by attaching assets and income that are not except from attachment. If the other driver has few assets you can take and doesn't have a large income it may prove to be more costly to get the judgment than you'd ever collect.
What you are entitled to by way of money damages when the other party is 100% at fault are the medical costs you incurred because of the accident (and if your medical insurance company paid any part of those costs, it will seek reimbursement for what it paid out), any property damage you sustained (again, your auto insurer would claim reimbursement of what it paid to you for that damage), and loss of income that resulted from being out of work due to the injuries you had.
In addition, you are entitled to some money for pain and suffering. Pain and suffering damages are notoriously hard to predict because different juries hearing the same case may come out with wildly varying amounts for pain and suffering. For settlement purposes, the standard practice in my state is to add in one-third of the personal injury damages for pain and suffering. A number of states use a similar standard, though I don't know what the common practice is in Arizona.
Your attorney will only want to take the case on a contingent fee basis if his/her analysis of the case indicates at least a reasonable possibility of getting a fee at least equal to the value of the lawyer's time if the case was billed hourly plus whatever costs the lawyer incurs that are not billed to the client. In many states, the typical contingent fee is one-third of what the law firm actually collects for the client. Even if the case is likely a winner on the merits, the lawyer won't take it if the amount of the damages won't be enough to make it worthwhile to do all the work that is involved, after taking into account the likely costs the lawyer will have.
Litigation can be expensive. The typical contingent fee agreement in my state is one-third of the total damages awarded up through the trial, and 40% of the total damages if the case ends going through the appeals process. You might be obligated under your fee agreement to pay some of the direct costs of litigation, like costs of depositions, discovery, etc out of your award in addition to the one-third the attorney takes. If the offer on the table now is anywhere to close to what you'd end up with after all the litigation costs are accounted for then the best decision is typically to take the offer and move on. Don't let your emotions take over your decision making process on this. It's understandable that you'd be angry at the at fault driver and want to exact your pound of flesh from the other driver and insurer to compensate you for what happened to you. But you need to keep your eye on what the probable end result may be and what risks you have that it may be lower than that and base your decision on that. The goal here is not to get the other party to pay as much as possible, the goal is for you to come out of this with as much money in your pocket as possible after all costs are accounted for.
If you think your lawyer's assessment may be too low, you are free to seek an opinion from one or two others. Ask each attorney you consult for the reasoning behind the recommendation so you can evaluate which attorney, if any, to hire.
