glmmTMB-Overdispersion in beta models on continous data

Question

I would like to model continous data ranging from 0 to 0.96 (the data is reflects the proportion of leaf damage on individual leaves).

I was thinking of using a beta regession since the beta distribution is a continuou probability distributions defined on the interval [0, 1]. See data distribution:

I modeled my data using the function "glmmTMB" using the following model structure (Chewin_propohn0= Leaf damage proportions)

nzmod3<-glmmTMB(Chewin_prop ~ Drought+Stratum+Year+Drought:Stratum+Drought:Year+Stratum:Year+(1|Tree.ID), data=ss,family=beta_family(link = "logit"), na.action = na.exclude)

The model runs fine and I do not get any error messages. However, when I check for overdispersion using the "testDispersion()" function from the DHARMa package I see that my model is overdispersed (see below)

My questions are: 1: Is overdispersion something that should concern me when using a beta-family model on continuous data (I could not find any clear answers to that question) (in some cases my overdispersion parameter is larger than 3) 2: If 1=YES, what alternatives do I have? How can I deal with this overdispersion I already tried a hurdle modelling approach (fit a Bernoulli model to the zero/non-zero data, then a conditional continuous model for the non-zero values) but that did not solve my problem. Thanks a lot Mike