So, for an OCXO that is spec'd at 5E-10/day, this does not look like what I would expect
I wouldn't say that. It looks pretty much exactly like what I'd expect. Your ADEV at 1-day intervals (86400 seconds) is right at 6E-10. Given the small number of samples, the error bars would plausibly reach 5E-10.
The oscillator could easily settle in at 5E-10/day or better with a bit more aging, depending on how long it's already been running.
and it is getting worse with age, like an order of magnitude?
There's definitely not enough information to draw that conclusion. What you are seeing with the upward trend to the right is just linear drift, which is occurring at a rate that's very typical for an undisciplined OCXO.
I can't seem to find any plots of OCXOs free-running for a week or more, but here's an 8-hour plot of a good-quality 10 MHz OCXO (HP 10811-60168) shown on the same graph as yours:

My measurement didn't run for anywhere near long enough to generate a stability reading at t=1 day, but you can easily see where its drift characteristics would have put it right in line with your OCXO if it had been allowed to run as long as your test did.
Conclusion is that your OCXO is probably working fine, and very close to its specs.