We took the plunge about 18 months ago, after having the same landline for over 25 years. I had pretty much switched everything to my cell phone, and my husband has a business landline in the house anyway although now he has started using his cell more and more. The functionality of a cell just can't be beat - once you wean yourself off the landline and the calendar on the fridge and a rolodex!
If you don't want a cell phone close to you head, then you use the speaker function and/or get a headset or set of ear buds with a microphone attached. I think some longer phone calls are not the problem - it's sleeping with the cell phone next to your head all the time, which some people do when they listen to music or use the cell phone as an alarm. But you can do that with it a few feet away from you with the phone on the night stand. I started keeping the cell phone next to my bed a few years ago when my elderly mother fell during a big northeast storm that caused the phone lines and power to go out for 5 days - her assisted living center called to leave a message, but our phones were out and I didn't hear a ring, obviously, or even get the message for a few days. Grrr….After that, I kept the cell phone with me all the time.
You get a few - very few - telemarketing calls on the cell, but I've been without a land line for all this time and have had maybe 20 of those calls. So it's definitely worth it.
As for the kids and their dad, you could consider a landline for just them (that you and Dad use) and which is severely restricted (like no long distance calls). To avoid the installation expense, talk to the phone company about giving you a new number on the existing wiring, and tell them it's for the kids and what kinds of limits can you put on that number not going out to telemarketers. Unlisted numbers are nothing new but you want to look at the costs involved.
Or you could get an extremely limited cell phone package that lets them call about 4 numbers (you, Dad, Grandma, the neighbor) plus always 911. No internet, no texting, no games that use up the data package, no nothing. They would have to keep it charged and keep it in a certain location of course - and if they don't, that's your perfect reason for showing them they aren't ready for their own cell phones anytime soon. If they ARE incredibly responsible with it, then you are training them for a few years down the road when it starts to look like a good idea/convenience to get them their own phones.
As for making appointments from the landline and filling out forms, why can't you do that from your cell phone? If you put it on speaker, it's not next to your head anyway - it's in front of your face a few feet away.
With the money you save by cutting off the landline, you could increase your minutes on your cell pretty easily. And you'd be rid of this big headache of unplugging, jumping up to see who's calling (and that's a habit that just has to be broken with the kids and their phone), etc. Reduce the number of rings too, so that things go to voice mail much sooner. A cell phone is so much easier to turn off and on than a landline. Just close it up or lay it face down so you don't see it light up every time someone calls or an alert comes in. And if you forget to plug your landline in for days at a time, then you are not really using it that much anyway. You're just attached to the security of it.
I think you'll find you can use the features of a cell phone much more efficiently if you work a little to learn what's there. And if you're watching a movie but waiting for a phone call, put the phone on vibrate and ONLY stop the movie to answer the call from that one person. Everyone else can learn to leave a message and wait for you to get back to them. 99% of those calls aren't urgent. You'll feel better if you prioritize better.
For charging in the event of a power failure, be sure you have a charger for the car - most cars have more than one plug-in for that - I have one in the console and an extra one under the glove compartment. With all the people who are plugging in GPS units and their IPhones for music, there are several ways to keep everything plugged in!
I also keep a charging cord in my laptop case. If I'm using my laptop at a library, coffee shop or any place where I have a business meeting, I just plug my phone into it, and it charges off the laptop battery or off the laptop charging cord (so I'm only using one electrical outlet).
Figure out a good way to keep your cell phone near you - special pocket in your purse, a belt clip, a wristlet style wallet that just holds phone/credit cards/license/paper money.
I think you'll enjoy the quiet!