It does not matter how you convey the message or try to explain the facts, YOUR insurance company will make that determination. Since you have filed under uninsured motorist, confirms that the other driver has no insurance, so only YOUR insurance company will determine if the other driver is at fault, if so, they will pay for your damages and go after the other driver for reimbursement of what they paid and a deductible you will most likely pay.
When you called your insurance company, the adjuster most likely had you give a recorded interview of what happened.
ie; speed, did you have a stop sign , did the other driver stop at the stop sign or just blew it like you said; were there witnesses, injuries, damages to both autos etc.
If as you said, they never stopped and struck you while crossing a street, then most likely you may or may not have had time to react or avoid their auto. But say you also had a stop sign, then the question is, did you stop and verify no ongoing traffic coming before crossing?
Don't worry what others tell you, since it will be YOUR insurance company that makes a decision of fault in this accident, since it will be your insurance company that will pay or not.
breathe, relax, most likely this will work out in your favor.
You can contact your insurance company and ask to include further details - like noting the Stop sign on the other road. Anyway, your original statement should have the name of the road you were travelling on and the road which the car at fault came out from. Ask to include that the other road had a Stop sign. If the insurer assessor does their homework, they'd find this out anyway, but best you tell them just in case.
If the road you were travelling on did not have any markings like Stop signs or Give Way signs, it would be understood that a vehicle coming out from the road with a Stop sign is required to stop and look for approaching traffic (ie, your car).
He who runs a stop sign is at fault for failure to yield the right of way.
Do you have a witness? That would help. Did the other person admit to running the stop sign? If so you're in the clear!
They're the one at fault.
There are 2 kinds of fault. One kind is for traffic violations. Cops and judges determine that kind of fault. The other kind of fault is to see who is liable for repairs to a vehicle. The involved insurance companies determine that kind of fault. No one here can determine either kind of fault. No opinion here will have the slightest effect on either kind of fault.
They are at fault for running through the stop sign, not you. As long as you were under limit and the other vehicle did in fact pull out without precaution then they are at fault not you. However, if you never exchanged information or at least got their plate number if they just ran away then there is nothing you can do. You should be covered if you have collision insurance and everything though.