Echoing various comments to ride your current bike for a month or so, leave the plump tires on-
their little extra weight is a good trainer. Most cyclists aren't happy to have headwind- that too helps with training, as noted above.
Latter half of sept thru early oct is usually a good time to get a bike- a cyclocross style bike is a great suggestion [i'm biased toward a Surly CrossCheck- it is so versatile and adaptable to what may be your changing needs- plus the ride is comfortable] Get a fitting with the handlebar level with seat - this will address your back concerns. Also swap out the cyclocross style tire with a touring/commuting tire - 32 mm Marathon is a good balance of features. I cannot agree that skinny eg 23-25 mm tires are a good choice for your current riding. A cyclocross bike will probably come with more of a racing saddle; swap it out for something more touring/commuting - a lot will suggest a Brooks B-17. They are a great saddle, I have three - but other saddles would be better at this time for you - the Serfas Rx or Terry Cite are good examples of what might work better, plus they are less $. At this time- stay with flat pedals; like MKS touring. Everyone probably will suggest going with clipless, IMHO resist that until next summer or a future bike. Commute by bike, if you can - even just once a week. Don't worry about distance, speed or time - just ride what you are comfortable. Cycling should be fun and a pleasure - so many are focused only on the sport aspect they forget the satisfaction of seeing and experiencing the world from a bike seat.