Sheila, I posted earlier but my post seems to have been swallowed up.
Although I have never held any stock myself, I have handled many executries (estates) that involved stocks and shares. I agree entirely with Judy, a good broker will really help you - maybe you have a friend who could recommend one? And I totally agree about finding one that acts independently and charges a fee - anyone who offers to do it for nothing will be raking off huge commission from recommending certain financial "products", and will not necessarily have your best interests at heart. Agree the fee upfront - any good broker will be happy to do this.
I used to do a lot of work on behalf of my clients with a stockbroking firm in Aberdeen, and I always found them extremely helpful and open in their dealings. Most of my clients did not want to have to manage the holdings themselves - in fact I think very few people do nowadays, it is far too complicated. The brokers I dealt with would manage your shares on either a "discretionary" or and "advisory" basis - one meant that they took all the decisions for you, but of course kept you informed, and the other meant that you instructed them and they simply carried out your instructions to buy, sell, or whatever. It is not just a question of buying/selling either, as even if you hold onto shares, things like "rights issues" crop up (these are when a company offers to sell more shares to its existing shareholders in order to raise cash), and a good broker can tell you whether or not these are a good deal (whereas the company itself will of course try to persuade you that it is). When my firm was a trustee of any estate (as lawyers often are) they always opted for the discretionary arrangement - there is no way that any non- finance professional would take the risk of making financial decisions for clients themselves these days.
I would caution against approaching banks, as in my experience they always want to sell you their products, which may not be the best thing for you. Similarly, some "wealth management firms" (usually the ones with thick writing paper and smart offices) will get you involved in very complicated investment plans that may not be appropriate, and which can be extremely hard to extricate yourself from.
You could of course just sell all of the stock, but you should still seek advice before doing so, so that you sell at the best time, and also so that you are aware of any tax implications there may be from selling. And if you do sell, you will need to think where you will place the sale proceeds - again, an independent financial adviser would probably be your best bet. Alternatively, you could just spend it all and have a good time! An old boss of mine was always telling people to do that - BUT she was wealthy, unmarried, had no children or other family - I think most of us with children feel that we should at least try to pass something down to the little dears, especially as it looks like they are going to be financially even more squeezed than we are.
Finally, I don't know if inheritance tax exists where you are, but if it does you may want to get some advice about that from the financial adviser. In the UK, if you leave more than the current "nil rate band" (I think it's about £325,000 at the moment but I am out of date), anything on top of that is taxed at 40%. (Gifts left to charities or to your spouse are exempt, but gifts to children are NOT). There are ways to plan your finances to reduce the exposure as much as possible, and the sooner these are thought about the better (in this country, even if you have given everything away before your death, the tax people can claw back tax from any gifts made up to 7 years earlier).
Sorry, that was a bit rambling, but I hope it helps a little.
Rosemary