Do your other campaigns all go to different pages as well?
While you probably won't have an even traffic split, you could create segments that look at the flow for Campaign A, Campaign B, Campaign C, and your new Campaign X, etc... or if you have too many campaigns to look a them all, you could do "all existing campaigns" vs the new Campaign X.
I would calculate out the % of traffic to your Campaign X vs the other campaigns (either individually or as a whole). Then yes, as you suggested, I would get stats for the comparisons based on visits, engagements, conversions, etc....
Then I would look at the various metrics against the traffic to each group.
For instance, let's look at Conversions....
If Campaign A brought in 50% of traffic, Campaign B brought in 20% of traffic, and the new Campaign X brought in 30% of traffic, then you look at your conversions... obviously conversions will be a subset of the traffic that came to each campaign.
You can obviously pull in conversions per campaign, but you can also do some math to see which has the best conversion rate (conversions over traffic).
You should be able to do other such calculations to see how your campaigns are working... and I'm sure once you start pulling some numbers, you will think of some new calculations to pull to see more in depth engagement.
Good Luck