I didn't quite understand where you're getting all your OH groups, maybe from the enol tuatomers of the nucleotides. I tried to look up the pKa's for the keto tautomers of the nucleotides on Wikipedia1,2,3,4.
In general, if pH is below a functional group's pKa, that group will be protonated, and if pH is above pKa, it will be deprotonated, though this is an equilibrium.
If we assume pH of 7 and assume I have the pKas right ( these values are for the free nucleic acids, no deoxyribose attached ), then the primary amines on A, G, and C will be protonated NH3 groups with positive charges.The amides on G and T should remain deprotonated and uncharged.
So with the sequence GGACT, we have 4 primary amines, so 4 positive charges. However, we also have 5 phosphates, each carrying a negative charge, so +4 plus -5 is -1.
Of course all of this goes out the window once we bind that oligonucleotide to another oligonucleotide, I assume the hydrogen bonding will alter the pKas and charges. For all I know, simply attaching the nucleic acids to a deoxyribose and stacking them together changes the pKas.