Within a country, a coin doesn't strictly need any backing at all - the reason that you want drachmai, in an economy where you can get by in trade without them, is because somebody will come from the state demanding taxes, and those taxes need to be paid in drachmai, and if you don't pay, you'll be punished. It doesn't actually matter what the coin is 'worth' as a material object, provided that the state is willing to accept it - if so, there's no reason for anyone else not to accept it, since it's just as useful to them.
However, there are two caveats to that. The first is that a debased coin might be counterfeit - somebody might have taken some genuine coins or the equivalent value of precious metal and melted them down with something else, making a profit - and the state will not accept counterfeit coins - in fact, it might punish their possession. The second is that 'international' trade doesn't work on that basis, since the Syrian merchant receiving your English shilling will never have to pay English taxes, and money isn't changing hands often enough to make it worth keeping it to trade with people who will. So in those circumstances, the 'exchange rates' are set by the weight of precious metal in the coin.
The problem with ancient debasement isn't, as often quoted, that it reduced the metal value of the coins and that this was necessarily tied to their exchange value, but rather that it was a means of increasing the money supply without increasing the amount of goods and services in the economy. Whether your money is on gold or on paper, that's inflationary.
What does decoration have to do with all that? Firstly, it is a stamp of authenticity - a signal that Offa will, indeed, accept the coin in taxes. By extension, it's a mark of purity - the very first coins actually had their centres punched, to prove that they were precious metal the whole way through. Why is this important? Because most people don't carry around weighing-scales, so the official mark of 'one dinar', 'one solidus', 'one drachma' or whatever is how they tell what the coin is worth. In this case, it's speaking the language of Arabic merchants (and those used to working in their currency), saying that this coin is equivalent to their own dinars - the implication being that they could weigh it if they liked, but there's no need to bother, as long as the coins are generally considered trustworthy. People also need some way to detect forgeries. The idea of the decoration is, then, also to be difficult to imitate - and here the Arabic writing would be quite a good tool for that, because it is at once complicated and easy to tell the difference if you're holding a 'good' coin and a 'bad' imitation against each other.