New to this and need help choosing integrated amp

I'm putting together a stereo system with speakers in 2 rooms: the living room and the kitchen. Next up for purchase is an integrated amp (then speakers) and I could really use your help.

The spaces: the living room is 13’ x 25’ with 11 foot ceiling, plaster walls, 5 windows, 2 sets of double doors, and carpet. The stereo will be on a rack in the LR with speaker wire to a set of speakers (TBD) in the living room and to a set of Energy C-100’s mounted high on a ceiling beam in the kitchen. The kitchen is 19 x 21 with 10’ ceilings and hard surfaces.

The system: in the last 2 weeks I bought a Project Debut III turntable, a NAD C-542 CD player & a Parasound Halo T-3 tuner. We have a Kenwood KA-3500 Integrated Amp my husband bought in college but it has problems. I thought my older Advent Prodigy speakers were broken but when the new equipment arrived I realized that the buzzing was coming from the amplifier, not the speakers. Deoxit didn't do much and so I need to replace the amp.

The music: we listen to rock (Springsteen, the Kinks, the Who, U2, Vampire Weekend) to folk-rock (Cowboy Junkies and a lot of Irish music w/ female voices) and to classical music (opera, Gorecki, Bach.) I have a collection of Jazz records and Mancini from the ‘60’s so I listen to those too. We also listen to the radio a lot – NPR & Canadian Broadcasting.

The need: first, we need to get a new amp. I’ve done some research and I’m torn between the Cambridge 640a + phono amp (about $580 total) or the Onkyo A-9555 ($440)or one of the NAD amps, used. Another option would be to return the beautiful Parasound tuner I bought ($400 and I haven't opened the box yet) and go with a receiver instead. I need a phono preamp and I need to switch from speakers in the kitchen to speakers in the living room. Sometimes I’d like all the speakers on at once (is that possible with these options?)

The need part 2: following this big decision I will need to replace the living room speakers, probably with 2 tower speakers and would welcome advice on that. I hope to spend less than $500 to $700 on the speakers. We need speakers that sound good throughout the living room, without one particular sweet spot.

The budget: it's tight because I am the wife and as you all know, there's a limit to what we'll spend on audio. I'd love to spend just another $1k on the amp/phono preamp and speakers, if that's possible.

By the way, this all started because my husband asked for a USB turntable for Christmas – one he’d seen at Costco for about $100. That released my pent up desire for a new stereo to fill our house with beautiful new music. Even with the humming amp, it’s been a treat to play records and listen to my favorite CD’s.
To your integrated amp question, I've owned the Cambridge 540, a couple of NAD integrateds (though not current models) and the Onkyo A-9555. The Onkyo is far and away my favorite in that group for sound, build quality and versatility. I think it's a remarkable bargain. There's a Stereophile review here. Accessories4less usually has refurbished units if you're interested in spending as little as possible.
You asked about the ability to play one or both sets of speakers with the amp. Yes, the Onkyo has A and B sets of speaker outputs and can play through either or both.

I'm not very familiar with tower speakers in your price range but some brands you might look at are Energy, Paradigm, PSB, Polk, Epos, and even Athena at the lower end.