Is there anyway to be able to access the landline phone signal over Ethernet of a LAN ... What I'm hoping to achieve is to be able to plug a handset into the switch in each room and be able to use it ...
No; as described, what you propose is not possible.
Any experiments you perform are at your own risk, at risk of damaging any equipment or cabling involved.
You might be able to use standard Ethernet (Cat 5 or Cat 6) cabling to carry telephone or audio-frequency signals, such as between wall outlet and phone or between phone and handset. This isn't what Ethernet cabling is optimized to do, so how well this works is anyone's guess. However, even if it works, it will not be Ethernet.
Since your Ethernet equipment, including switches, assume that what is being carried over the cables is Ethernet, with the specific electrical and protocol properties of Ethernet, what you are proposing won't work. At best, it just won't work; more likely is that the equipment may be damaged.
Just use the proper equipment for each job. It will work much better.