In my code, I create a TxDb object from makeTxDbFromEnsembl() call. Every time I run the code, it takes quite a long time - connecting to ensembl server, fetching the data, etc. - even though the data on the server did not change. Is there any way to avoid this by saving the database locally and then only checking that the current database release (i.e. the string like homo_sapiens_core_99_38) is the same?

Yes, take a look at the "Saving and Loading a TxDb Object" section of the GenomicFeatures vignette.

There is a pair of saveDb() and loadDb() functions that do exactly what you want.