Starting from "copulas" may not be a natural way of doing this...where "copulas" is interpreted as making the joint distributions important. It may be more natural to start from the forward-in-time conditional distributions, and to let the joint distributions fall out of these (by simulation if necessary.

However, taking this "conditional" approach may lead you to wanting to bring into account other variables (such as the amount of rainfall between drought events), which may be going beyond what you are thinking of.

Note also that, thinking that the forward-in-time conditional distributions are important may have implications about exchangeability in the joint distributions. That is, the forward-in-time conditional distributions are not necessarily the same as the backward-in-time conditional distributions.

If you already have copulas to describe variables within a event, the most direct way of moving on to consider dependence between events would be to investigate how these copulas change depending on the past ,for example by dividing "the past" into two classes "high" and "low".

In your original post you said "as the droughts are generally correlated" ... the suggestion of using high and low classes was as a starting point for exploring this statistical dependence. So, since you mentioned severity and duration you could use either of these (or both, in separate analyses) as the basis of splitting the data-set into two parts (based on the past) and then trying to characterise the change in distribution between the two parts.

I may have jumped to the assumption that you have and are familiar with software based on fitting copulas and so could use this. Perhaps you are thinking of "copulas" as being equivalent to bivariate distributions, but copulas are a specialised version only applicable to some continuous distributions. If your case is such as to have only a few different values in one or other variable you may need to look elsewhere. For a simple graph-based exploration you might try "convex hull layers" for a pictorial analysis of a joint distribution. An internet search should produce some examples ... you would probably need care to find a version of code suited to discrete-valued data.

Unfortunately I have no experience in drought analyses, so can be no real help beyond suggesting an internet search such as https://scholar.google.co.uk/scholar?q=drought+events+dependence&hl=en&as_sdt=0&as_vis=1&oi=scholart

You might want to seek out help from a local statistician for any really detailed analyses.