Yours is sort of a two-part problem. The first problem is that the bacteria needed to convert ammonia into nitrite does not do well at a pH below 7. If your pH stays at 6, you are going to find it very difficult to establish and maintain a stable cycle.
The second part of the problem is, at a pH of 6 most of your ammonia is in the form of ammonium, which is much less toxic. This will still show a reading on your test kit, but it is not as harmful towards fish and other livestock. However, if you raise your pH over 7, all this ammonium converts back to ammonia which is toxic as you know, and will possibly poison your fish.
This is made doubly difficult by the fact your tap water has ammonia in it already.
I agree that you should invest in some Seachem Prime. This will detoxify ammonia (turns it into ammonium) so it can still be used by your bacteria and will still be picked up by your test kit, but as I mentioned before will not harm your fish unless it reaches a very high amount. Hopefully, if your tank cycles it will be able to process the ammonia in your tap water. However, at present it may be best to either use a different source of water if you want to cycle your tank, or keep your pH below 7 to avoid poisoning your fish and dosing with Prime.
If your water is particularly soft (a low KH value) it will not take much for your driftwood to bring down the pH. KH is very important when trying to increase or decrease your pH as it is the main driving force behind it.