I speak Malayalam, one of the Indian languages and also Hindi but there are always common words which I assume are original Sanskrit words? So are the languages truly distinct or can I say they are similar dialects? What would be the apt technical word representing such a situation?
There are hundreds of distinct languages in India, from two main families, Dravidian and Indo-European. Malayalam is Dravidian and Hindi is Indo-European, so they are not only different languages but unrelated ones. All the major languages of India have borrowed extensively from Sanskrit, hence the lexical similarities you mention; but this doesn't make them dialects of the same language, any more than the fact that the Hebrew for "telephone" is telefon makes Hebrew a dialect of English.
It may moreover be noted that Malayalam could be considered a dialect of Tamil up to somewhere between the 8th and 11th centuries. From then onwards it went its own way and was heavily - and by far more than Tamil - influenced by Sanskrit, which may be seen by the mere fact, that the Malayalam script incorporates many non-dravidian Sanskrit sounds. Whereas there is only one labial (dental, retroflex, palatal, velar) plosive indicated by the tamil script, there are four of each of them in Malayalam, much like in the Sanskrit script. This is reckoned to be due to the influx of Nambudiri-brahmins from northern India from the 8th century onwards.
Broadly speaking, there is a linguistic continuum across Northern India. This means that there are no hard borders between Sindhi, Panjabi, Gujarati, Marathi, Hindi, Bengali, Oriya; instead there is a gradual transition from dialects of one to dialects of another. The Dravidian, Munda and Khasi languages do not participate in this continuum, nor does Sinhalese, which, though Indo-Aryan, is separated geographically from the other Indo-Aryan languages.
The apt technical term for formerly different languages developing common features is sprachbund. And yes, there is a Sprachbund on the Indic subcontinent.