Impact evaluation of a policy when different individuals adopt the policy at different times
The description is bit vague  on how many organizations we are talking but you could still do differences-in-differences (DiD) for all treatment-control-pairs that adopted it at the same time  separately and then running meta-analysis on the results.
As far as I understand based on Angrist &amp; Pischke (2009) Mostly Harmless Econometrics ch.5, DiD solves any possible self-selection bias as long as the identifying assumption of common trend is satisfied. That is in DiD it it is required for both treatment and control to follow common trend pre-treatment. If the changes in outcome variable ($y$) pre-treatment are virtually same between treatment and control, then even if the levels of $y$ are different the estimates won't be biased. This of course will have to be tested in each case.
The above will however only be feasible if you do not have to run too many of the individual DiD otherwise it would be unpractical.
An alternative could be the event study design (see MacKinlay, 1997) that is used quite often across finance literature. It allows for events to occur at different times and it does not matter that firms/organizations experiencing the event are heterogenous and possibly self-select for the event (this approach is often used to evaluate M&amp;A effect on stock returns for example), but on the other side this approach is very data intensive and depending what precisely is $y$ it might not be appropriate, or also if there is event clustering.
In order to implement regression discontinuity design (RDD) there must be some cut-off point that determines whether treatment is assigned to subject or not. For example, RDD would be appropriate if there is a policy that requires any power plant producing more than let's say 1.21 gigawatts is required to adopt some $CO_2$ curbing technology and below that threshold it is not required, and you want to examine the effect of such policy on some outcome variable like air quality. There is also so called fuzzy RDD which does not require sharp cut-off in some deterministic variable as long as probability of adopting the treatment is significantly different  (see ibid. (2009) Mostly Harmless Econometrics ch.6). The feasibility of this will again depend on data availability and other details, but if the organization themselves have full control over probability of treatment it will not be appropriate.
Lastly, as you could have probably already guessed from the discussion above the answer the choice of methodology is always context dependent. In order for anyone to give you concrete answer you would have to share virtually every single detail of the study which is not feasible, so take the above as some pointers with references (since you tagged the question as a reference request) rather than telling you exactly what to do.