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DearCustomer- I see no relevance between owning stock in a public company and getting a name transferred on an electric bill. In order to transfer the stock you will have to open an estate for your husband. If there was a will and you are the executor then you can handle everything according to the will. If there is no will then you can apply to be the administrator and distribute the estate according to intestate succession.
As far as the electric bill being transferred, if the property is in your husband's name you will probably have to get that into your name through the estate process or by virtue of a joint deed with right to survivorship before the power company will change the bill. There's really no hurry to do this since, if the bill is paid I'm quite certain the power company doesn't care whose name the bill is in.
Your credit may come into play at some point but the only thing that may be required is a deposit being made to the power company. As long as the bill is paid they can't shut the electric off.