A will is only valid if the testator's signature is witnessed by two people. If the witnesses to the handwritten will are still alive (and can be contacted) that would seem to be the obvious way to check that the testator's signature is genuine.
The executors of a will don't acquire their powers to distribute an estate until they've received grant of probate. Unless the estate is unusually complicated there's absolutely no reason to employ a solicitor to seek probate (unless, of course, an executor is actually a solicitor anyway). It's a really easy process, which just involves a bit of form-filling and swearing an oath.
Chris