The alarm that won't switch off
Before changing anything, it helps to understand what is actually happening when a late reply ruins your afternoon. It is not that you are needy. It is a system doing its job too well.
You have an attachment system, and yours is set sensitive
Everyone has an attachment system: a very old part of the brain whose only job is to keep you close to the people you depend on. When it senses distance, it raises an alarm, and the alarm makes you want to close the gap.
In some people the alarm is set sensitive. A slow reply, a flat tone, a change of plans, and it fires, flooding you with the urge to check, ask, fix or pull closer. That is what attachment anxiety is. Not a flaw in your character. A smoke detector that goes off when someone makes toast.
The goal of these four weeks is not to switch the alarm off. You need it. The goal is to recalibrate it, so it goes off for fires and not for toast, and to give you something better to do when it does.
John Bowlby described the attachment system in the 1950s and 60s. Later research by Mario Mikulincer and Phillip Shaver showed that anxiously attached adults use "hyperactivating" strategies: they turn the alarm up to make sure they are noticed. It works in the short term and costs a great deal in the long term.
Turning the volume down, first
Breathe in through your nose for four.
When did the alarm last go off? Describe what set it off, as plainly as you can.
Quick check
Your partner replies to your message three hours later than usual, with "sorry, crazy day". What is the alarm most likely responding to?
Take this with you
The alarm is real. It is just not always right. Those are two different things.