Should I perform both lemmatization and stemming?

I am building a classification model. I am performing the following processing on the tokens:
Basically first I lemmatize the word and then stem it using python libraries.

The reason I don’t want to just lemmatize is because I noticed that WordNetLemmatizer wasn’t handling some common inflections. In the case of adverbs, for example, lem.lemmatize('walking') returns walking .

Is it wise to perform both stemming and lemmatization? Or is it useless?

I think stemming a lemmatized word is redundant if you get the same result than just stemming it (which is the result I expect). Nevertheless, the decision between stemmer and lemmatizer depends on your requirement. In some cases stemming increases recall and lowers precision and the opposite for a lemmatization.
Consider these scores and the impact of this processing on them and then decide. The other way is to calculate F-1 score which is the harmonic average of the precision and recall and then compare between the scores for both stemmatized text version and lemmatized text version. This will give you a fair idea about what is the best for your data.