Yes, to match sense 7a you need the with1. That definition is for a particular use of the word which has the elements
[container-type thing or space] [crowded with] [individual things or people]
Those pieces can be arranged in slightly different ways (and the verb can be conjugated differently, of course), but you must have those pieces in that relation to one another: the crowdedness describes the state of the container-thing.
The trash crowds in the bin.
puts these elements in a different relationship with one another. Instead of the bin (container-thing) being the subject and describing it as crowded with the [pieces of] trash (individual things), in your sentence the trash is the subject, and you describe it as crowding in the bin. Here, the crowdedness describes the state of the trash, not the state of the bin.
But don't despair! Your sentence is fine, it just matches a slightly different definition, the OED's sense 6b:
To compress; to collect, bring, or pack closely together, as in a crowd.
And actually matches fairly closely the 1776 quotation for that sense, "flowers crowded, in the bosom of the leaf-stalks."2
As an aside, the OED is primarily a descriptive dictionary, not a prescriptive dictionary. So I would tend to ask "am I using this word in this OED sense?" rather than "can I use this word in this OED sense?"
1 You could also probably use by, but you haven't asked about that and I think with is usually going to be the better choice.
2 The comma looks superfluous there to my eye; I think in modern writing we would not include it, making this a near-perfect analogue for your sentence.