This sounds a lot like an arp cache issue although the fact when you switched back they still didn't work doesn't quite fit.
Basically when you have static NATs the firewall uses proxy arp.
So the IPs are all in the same subnet and an IP from this subnet is also on the upstream device ie. either your or the ISP router.
When the upstream router asks for the mac address for one of those IPs your firewall responds with it's outside interface mac address so the router will forward the packets to the firewall.
The router will keep a record of this mac to IP mapping in it's arp cache. When you switched over to the ASA obviously that has a different mac address on the outside interface but the router doesn't know this.
Until the router times out the arp entries it will be sending packets to those IPs with the wrong mac address.
The reason you could get to the outside interface IP is because traffic is coming from inside to outside and I'm assuming you NAT internal client IPs to the outside interface IP so this would automatically update the routers arp cache.
But obviously that wouldn't happen for the static NAT IPs because the traffic is coming from outside to inside.
The only thing that doesn't fit with this is why when you switched back you had to change the IP before the static NATs started working again because the router would have the right entries.
So either I'm missing something obvious (it's a bit late here and I'm tired :-)), or this isn't the issue.
The solution is to either do this out of hours ie. after everyone has left and then hopefully the entries on the upstream router will have timed out by the morning or -
a) if you control the router clear the arp cache
b) if you don't you need to liase with the ISP
Like I say, it may not be the issue but it does sound the most likely reason.
Jon
