How do i change the message a caller hears when the phone is not working after business hours?

Yes - Afraid that is the case, certainly in the uk.
Phone will only accept a message if phone is not in use / not answered.
You do get notified that there is a call waiting so you can either switch to their call and ask them to call back.
Or when you have finished call - their number will show as last caller and you can call them back.

Try greetings 2, 3, or 4. One of these might be the after hours greeting.

mY dear Your Phone Ringer Default So Please change Ringer
And Ringing Sound Problem will solved

When you are on the phone the caller will get a busy signal, unless you have call waiting on the line, then the caller will hear ringback. If you have call waiting you can answer the second call by sending a flash signal to the service provider which puts the first call on hold while you are connected to the second call but your answer machine cannot answer the second call because you have only one line therefore only one speech path. So the answer to your question is no. If you have the voicemail feature from your service provider then the caller could leave a message not on your answer machine but in your service provider mailbox.

on the system you use the vm msg button and your password, you also need to know which numbers your messages greetings, hold, holiday are set at then that is the msg you change.
vmmsg
*# number changed
password
1
number you are changing
3
*
#

assigning messages is done by
feature 983>1020000>AA>Table>table 1>Greeting Type
depending on how you have the business hours times set it'll play the greeting you want. Typically I set the greeting type as morning:1, afternoon:1, Evening:1 and Non-business:4. then under business hours I set them up as mon- fri morn:0800am, aft:0800am, eve:0800am and nonb: 0500pm. sat and sun as morn:1200am, aft: 1200am, eve:1200am, and nonb: 1159am
press Rls when finished
that way when a caller calls between 8am and 4:59pm they get the day greeting and from 5pm to 7:59am they get the night greeting (greeting 4) saturday and sunday play the night greeting all day.
Dont forget to record greeting 4 as your night greeting.

Headset says ringer off, cannot hear dial tone

Sounds like the answering machine is picking up the telephone line and not releasing it.

Hence the busy signal, cause the answering machine has the line off hook. It's like leaving your telephone off hook, the next caller will get a busy signal...

The answering machine might be bad... have you noticed that you were talking on the phone and hung up then the next call would be the busy signal? Maybe test this out.
Do you have the correct voltage to the answering machine, too much power or too little power can do funny things to telephone equipement.

You are asking about changing you voice mail and phone system settings. Your phone can work on several different Panasonic phone systems and you could have different voice mail systems as well.

 

The phone KX-T7633 has only a user guide that covers the basic features of the phone itself. You need a programming and installation manual for your phone system AND your voice mail. They are not the same part and have different manuals.

 

So, your first step is to find out the model number of your phone system and voice mail system.

You may have activated a "Caller tune" or "Ring back tone" service. Check with your service provider to deactivate the tune.