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My understanding, original 3560/3750 licensing, was, as you note, the -E suffix should have licensed the device for IP Services. Way back, there was also an Advanced IP Services IOS, needing another license, but Cisco later lumped the two Services versions together under one license and just one IOS image.
If you have maintenance on the device, you're entitled to any subsequent versions with the same licensed feature sets. However, the original 3560 series is well past EoL, so maintenance isn't available for it, I believe. So, that's one wrinkle. Another, I recall the original 3560/3750 series also originally had a lifetime software version upgrade license. (This, I think, was to compete in the SMB market, where such was common for other vendors.) I.e. as long as you owned the device, you could upgrade the IOS, of course, again keeping to the same feature set. I further recall, though, Cisco may have revoked that. I.e. later you needed an active service contract for any IOS version upgrades (much as typically done for all their other Enterprise devices).
The last wrinkle, I think all IOS licensing only applies to the original buyer. I.e. from Cisco's perspective, unless you purchased authorized refurbished units, you don't have a legal IOS to run on them. Personally, as long as units were legally acquired, you can probably "safely" use them with whatever IOS was installed on them, at time of purchase, assuming the installed IOS was legally installed before you purchased it.
That said, Cisco could, and probably would, say as a 2nd hand buyer, your cannot obtain any IOS upgrade. Legally, I think they are within their rights to do that.
To recap, as worst case, you've purchased devices you're not licensed to run any Cisco IOS on, unless you get their approval. Best case, if you contact Cisco, you can work out an agreement to upgrade the IOS versions. Probable case, you're likely (legally) "stuck" with only running the IOS that was already installed.
