Should I back out of this job offer?

Should I back out of this job offer?

I'm in the middle of a painfully long 3 month notice period with my current company and have a new contract that I signed back in March. (I haven't told the old job about the new job.) I decided to leave because of big time mismanagement and overwork that led to burnout.

I'm feeling like I want to back out of the offer I accepted because they haven't communicated with me much since mid March when I signed the contract. I emailed HR asking a question and got kinda rudely redirected to only talk to my recruiter until I start. And wrote her a pleasant email with a doc I needed to share and saw that she CC:ed me into passing it along, but no personal reply. Am I being too sensitive thinking that even checking in when I don't start until July is just good protocol?

The old job is spinning my team into another agency they acquired and the guys behind it are desperate to keep me. But they're extremely disorganized and I'm not sure it would be anything but more stress. I ended up casually interviewing with another place that also wants to offer me a job, but they can't go as high as the current offer I have. Even if it's neither of these in the end, I feel like the least a new company can do is be welcoming over this long period just to keep me engaged. Thoughts?