Welcome to Wikivorce, although Im sorry you find yourself here.
Adultery is consensual sexual intercourse between a spouse and someone
else they are
not married to. However,
proving that
adultery has taken place is very difficult.
If you are simply dating, and the relationship hasn't moved on to be a
sexual one, then no you aren't committing
adultery. Might not stop her claiming that you are,
though

You have only been separated a very short
time, and while I can understand the need to "move on", a
separation/divorce where the spouses both have new partners can get a bit
messy and complicated.
If you wish to divorce her, I would
suggest you use Unreasonable Behaviour, citing an inappropriate
relationship with a third party as one of your examples, rather than using
adultery on the
petition, because if she denies
adultery, then the
petition won't stand in court, and you are back at
square one.