If I was in a car accident because the other guy ran stop sign and tore my right shoulder rotator cup which required surgery to fix, what is a fair settlement?

It is impossible to answer your question without knowing all your circumstances.  In general, if the liability of the at-fault driver is very clear and undisputed, you have a strong claim.

The amount of settlement depends on your injuries.  First, you should make sure you are fully healed before you settle a claim.  Make sure you will not need further surgery.  Make sure you have good  or perfect function of your shoulder and arm.  Only then should you consider a settlement.

If you settle before you are fully healed, you will not be able to come back later and sue for more if you don't recover well or you experience complications.

Also, understand that the insurance company will demand repayment for any medical expenses.  A disability insurance company, or perhaps your employer, will also demand repayment for disability benefits.

A fair settlement should pay all of your medical expenses and repay all disability benefits.  It should compensate you for any lost wages in the past and in the future.  After that, you have to consider how your life has been altered by this injury.  Will you undergo rehab for 6 months and then be perfect?  If so, compensation is less.  Will you suffer for a year, have a permanent disability, and then have to find a new profession, trade, or line of work?  If so, your compensation will be quite significant.

You also have to consider how much insurance the at-fault driver has.  If he/she only has $25,000 in coverage, the company should pay all of it, right now.  If he/she has $100,000 in coverage, they would probably be responsible for significantly less than that.

Are you in Cleveland or a rural area?  That matters, too.  Juries in some areas of Ohio traditionally award more money than juries in other areas.  What is your background and employment?  This should not matter, but it does.  Employed people with families and good jobs get more money than people without these circumstances.  Felony convictions are sometimes admissible in evidence even if they have no relevance to what happened or your injuries.  If you have felony convictions in your background, that is going to decrease the amount of your settlement.  I wish none of this was true and settlements were based only on the injuries you suffered.  However, that is not reality. 

My bottom line advice is this -- if the at-fault driver has insurance, and his or her insurance company is offering you the full amount of coverage minus 1/3 (1/3 is about what you have to pay an attorney), you should seriously consider taking it (unless the driver is rich and has assets).  If this is not the case, you need to hire a lawyer.  You have a serious injury and you will recover more with a lawyer than without.  The simple truth is insurance companies take lawyers seriously and rarely take unrepresented people seriously.