I would go either or or rather one after the other. That way you can focus on the important part of a scene and get the most out of it.
As always, I like to look at it through the lens of action movies. Often times, you've got a chase then the fight, or the fight turns into a chase. During a chase, people might shoot, but very rarely does the chase end because of a gunshot, they are either part of the background noises (albeit important for dramatic effect), or they are used to shoot windows, etc., also for dramatic effect.
So using the guns skill in a chase can easily be done, either by doing it as a maneuver for an attempt at a +2, or as the skill roll for one exchange, where you focus on intimidating the other by shooting at him.
On the other hand, if you have a fight even during a chase, the protagonists usually are fairly static towards each other. The fight often ends by them being forced apart, which can lead to a chase, given the right circumstances.
In this case, you can use the Drive skill for maneuvers or sometimes even to defend or to attack, as long as you can justify it by what's going on.
Setting the two scenes up can be done by having one be the outcome of the other. So if you chase someone, success would mean that you can now fight them and take them out. If you are chased, failure could mean you are forced to fight your pursuers. And so on.