this.
I wouldn't even consider overtime until you complete an initial assessment in terms of hardware warranty, redundancy and replacement costs, licensing mitigation etc. For that, you hardly have to get under the hood.
If they open their budget and agree to realistic (meaning expensive) hardware and software fixes which you'll surely need, stick around, otherwise forget it. Your resume is still warm, keep it in circulation. You don't want to be the lone IT guy in a shop full of illegal copies of server software.
If they magically agree to opening the purse, backup/restore and disaster recovery would be my first priority. Spend time figuring those systems out, implementing automatic tests and work in parallel on a hardware analysis of the entire place, power first, RAID second, database replication a close third. That is, as you review a hosts' backup requirements, and test restores, run through the host and check hardware, ensure it's appropriate to the task at hand, legal, secure, stable and all that. But don't dwell on any particular host too long at first. You don't want to get target fixated on one little detail for more than a few seconds.
Anything without warranty that doesn't have spare parts on hand gets replaced, anything critical without internal hardware redundancy (multiple power supplies on different circuits, RAID etc) gets replaced.
While you're ordering hardware, stage the deliveries so they can be replaced at a reasonable pace, and document as you learn. Start with a simply brain dump of the business reason a host/process exists, then dig down to the details of the changes from a default install of an OS while you're working to replace it.
I would insist on a PFY too.