I don't necessarily take issue with the five-day separation between the two cross countries, but mostly I advise some good, hard thought because I don't think you realize how hard this trip will be. Flying solo is a lot different than flying with a CFI, especially in the cross-country environment.
Even if you think you were making all the decisions, doing all the weather recognition and avoidance, and flying the airplane for that 15 hour cross country flight last weekend, having the instructor in the right seat takes a lot of the cognitive burden off your shoulders. You know that if you make a bad decision, the CFI will correct it for you. It may not seem like a lot, but it adds a whole new level of stress to cross country flight when you are solo and alone, especially as a new solo student (which, with only five hours, you are).
When I was training, my CFI trusted my judgment and by the time I was soloing, she was more comfortable with my abilities than I was. She probably would have signed me off for a flight like the one you're proposing without issues, whereas I would have been a lot less blasé about it. A big part of my caution was because solo cross-country was much more exhausting than I was anticipating, and the fact that you have no solo cross-country time is worrying since you don't know if or how it will actually impact you.
My long solo cross-country was almost exactly the same length as the distance proposed, and when I got back to my home field, I was so tired that my decision making was significantly worse than usual. There were two runways, one paved and one grass, at my home airport, and they crossed at almost exactly 90*. I got the weather when I was about ten miles out, and the winds -which were forecasted to be calm/minimal when I took off from my last landing spot - had changed to be 9G15, almost 90* from the calm wind runway that I was planning on using when I took off. Instead of changing to the runway that the wind was nearly straight down, which I would have normally done without an issue, it never crossed my mind that I should switch runways. I just stuck with the original plan and executed a safe, but significantly less than pretty, crosswind landing when I could have and should have done a much simpler landing on the crossing runway.
That landing not only was a stupid decision, it also busted the solo limitations for winds that I had at that time. Normally, I was very cognizant of keeping within the limitations. Switching runways based on prevailing winds was even something that I had done frequently before at this point - I would go practice landings in the mornings in 2 hour blocks, which often resulted in playing runway hopping to keep the crosswind component under control as the winds changed. I got lucky, because I did have the skills to deal with the crosswind as my CFI and I had gone up under very gusty winds a couple of times to practice crosswinds, but if I was not so tired, I wouldn't have needed those skills to keep myself and the plane together in the first place. If you would have asked me if I would do something that dumb before I left, I would have assured you there was no way something so stupid as a landing like that would be how I ended my cross country.
Sorry for the long story, but I don't want you to fly to an airport that you are not that familiar with and end up doing something stupid or pranging the plane because you didn't account for the human factor well enough. That's not to say that you aren't capable of this flight, but I would just be very careful to ensure that nothing induces "get-there-itis" on either the trip there or back as inexperience combined with a need to be somewhere is frequently deadly.
