Don't assume you are being attacked as Zone alarm will indicate at all activitys including legitimate ones such as port scans which your ISP operates to ensure that you are not abusing their services by running a server os similar, IRC and many other services also do legitimate scans....
Zone alarm can not correctly define the difference between legitimate and non-legitimate services so makes you aware of everything.... The wireless livebox you had previously very likely was a NAT firewalled protected router which due to its abilitys would have stopped zone alarm from making many of these alerts where as a generic modem will not...

The thing to do is read the firewall logs and establish what the alert means and what port is it using before assuming that a change of IP will resolve anything... it could be a ligitimate service scan, if you need help doing this post the logs Zone Alarm creates

Zone alarm is a POS imho that looks like its doing something amazing by alerting the user to all activity including those that the user does not need to care about, it also does little to educate about what is actually happening and how it effects the system, it makes claims about making the system 'stealth' as if this is important to security, which it is not
Zone Alarm has very little effective packet filtering as well.

Your ISP also provides a dynamic IP that releases and renews itself at a given point in time, this is to make it harder for users to run services such as webhosting which requires a static IP to be really effective (although you can get round this). The dynamic ip service means you do not need to ask for a new one as this is provided to you without knowledge of the fact... Freeserve renewed the users IP address twice a week, no idea if this has changed since it became Wanadoo, Orange or whatever its called this week