Both--GROCERY and GROSHERY--are correct. There is always another way of pronouncing the same word.
Lexico gives the transcription /ˈɡrəʊs(ə)ri/ (GROCERY) for "grocery". Is there anything in the transcription that draws attention?
In "casual" speech, unstressed schwas are usually chewed off; /ˈɡrəʊs(ə)ri/ → /ˈɡrəʊsri/ -- I removed the schwa.
Now there's another thing that catches my eye. Does it catch your eye too? I suppose not.
It is the /sr/. The consonant /s/ is called an alveolar sibilant; it's produced at the alveolar ridge--the bony ridge behind the top teeth--and is characterised by a hissing sound. The /r/, on the other hand, is a post-alveolar approximant.
In casual speech, we make the sounds that are next to each other more similar to each other (aka assimilation, which is a very common process). The /s/ is originally an alveolar sound, but when it's followed by a post-alveolar sound, we tend to make it similar to the post-alveolar sound. Assimilation can be described by this example: this shoes - how do you pronounce this phrase? In casual speech, it is nearly always [ðɪʃʃuːz] rather than [ðɪsʃuːz]. The /s/ changes its place of articulation because of the following /ʃ/ to make it easier.
When the /r/ comes right after the /s/, the /s/ becomes more like the /r/. In other words, the /s/ changes its place of articulation--which is alveolar--to post-alveolar (back of the alveolar ridge). In post-alveolar region, we have another sibilant /ʃ/ (the SH sound), so the /s/ changes its place of articulation from alveolar to post-alveolar and becomes /ʃ/. That's why you pronounce it [ˈgrəʊʃɹi] (GROSHERY) in fast speech.
The word "issue" used to be pronounced [ˈɪsjuː] - IS-YOU in Britain (most Brits still pronounce it that way!), but due to the same phenomenon, people pronounce it [ˈɪʃuː] - ISHOO.
It is surprising that this assimilation only happens in "grocery" and not other similar words like "nursery", "serene", "sorcery" etc. Maybe it will happen in those words one day.
Hope that helps!