I've read that some native speakers say the "r" sound by bunching their tongue (bunched "r") and some by pointing their tip at the alveolar ridge (retroflex "r"). When I say it, it sometimes feels more natural to pronounce it one way, and sometimes the other, depending on the sounds around it. Do native speakers also do this? Should I practice only saying it one way?
Yes. According to Mielke, Baker & Archangeli (2010), in American English, some speakers consistently use only one shape, some others employ both interchangeably, and others alternate the two consistently depending on environment.
Consonants that involve the tongue body (especially [ʃ] and [k]) and [i] are more or less bunched already, so /r/ adjacent to these sounds tends to be bunched. Retroflex /r/ is common when next to labials, word boundaries, and back vowels because these sounds do not interfere with retroflexion.
As seen in Zhou et al. (2007), however, there are various degrees of bunching/retroflexion and the shape of /r/ can often be somewhere in-between.
As for your practice, you need not worry or be conscious about it so long as you can produce either of them because there is very little audible difference. You should use whichever you find easier to pronounce; in fact that's what native speakers do.