- Who vs whom - Yes, this is picky, but, strictly speaking: "The man to whom I was speaking" should be used instead of "The man who I was speaking to".
- "James and I went to the shops" is much, much better than "James and me went to the shops".
- "Awhile" used incorrectly: it means 'for a short time', so should never be preceded by a preposition. If you want a preposition before it, you have to use the two-word form 'a while'. Conversely, "stay a while" is incorrect1, while "stay for a while" and "stay awhile" are correct.
- 'That' being used as a relative pronoun when 'which' is much better. Microsoft's grammar checkers suggest this because it makes things more readable.
- 'Grammer' - yes it's a spelling mistake, but it's appropriate, don't you think?
1. TenMinJoe and dictionary.com think this is correct. Looking at it again, they might be right.