I am new to the Mooshi unit and do know know as much about the unit as I would like, but it is doubtful the meter by itself could measure ESR. Usually to measure capacitance you would need some form of compairitor circuit as I recall. I am sure some more knowledgeable folks may be able to give a more concrete answer to your question, but I am not seeing this as a standard, stand alone capability of this type of product??
As for a decent Capacitance/EWR meter, I would suggest the Peak ESR70. I have had one for a while and use it quite often and I have been very happy with it. You can shop it around, it is usually just over $100 as I recall, this appears to be the cheapest I found with a quick search – https://www.vellemanstore.com/en/velleman-esr70-capacitance-and-esr-meter?utm_source=google_shopping&m=Abstract&gclid=Cj0KEQiA6_TBBRDInaPjhcelt5oBEiQApPeTFy7-AuDfIetN7SE5oWbk5wBNBb1YpXP8Zq26YNtYzkIaAkE28P8HAQ