Please help ...integrated amp under $1200

I agree with those who recommend you buy the speakers first before any other component. If you don't, you are playing russian roulette with your audio system.

With speakers there is something within the specs that is called sensitivity. 
Sensitivity is a measure of the relationship between the power input and the sound output. The standard for achieving this rating is 1 watt/1 meter. This means the speaker will produce sound at the volume of xx dB, measured by a microphone placed 1 meter away when it's given the input power of 1 watt. Basically it means the higher the sensitivity the better.

Another important factor to think about is frequency response. The range of human hearing is generally regarded as being from 20Hz, very low bass tones, through 20kHz (20,000Hz), the very highest treble. Presumably a speaker that could reproduce that range would sound lifelike. Basically, this means the lower the first number and higher the second number. However, this is not always true. You need to consider how the speakers are constructed and the components they use in building the speaker.

For example, I have a pair of mini tower speakers with a sensitivity of 92dB and a frequency response of 48Hz - 24kHz, and a pair of bookshelf speakers on their stands with a sensitivity of 90dB and a frequency response of 55Hz-28kHz. I have a solid state and tube integrated amp, and the second pair of speakers sounds better in my system in my room which is 11ft wide x 14ft long x 8ft high.

Do yourself and your wallet a favor, purchase the speakers before any other component. You will be more happy doing this.
 

If you only want to buy new, the Marantz PM-8005 is exactly $1,200.  Has excellent phono amp and headphone amp.  They can also be picked up refurbished from several dealers for $999 and used for $700-$800.  This amp is hard to beat for the money.

I second checking out a Marantz 8005.  I got one two years ago as backup to my Vintage McIntosh separates. I was curious to hear a modern integrated and auditioned Rogue, Marantz, Rega, Cambridge,NAD, Yamaha, others. I was very taken by the Marantz because it offered good clean dynamic sound, transparency, a neutral to warmish sound, ample current, direct to amp signal or extensive tone controls, ability to use two sets of speakers, preamp out etc. I got it and have set it up as a second modern system right beside my semi vintage kit. It was made in the Japanese Marantz  Reference factory sounds very close to their Refernce gear (I compared) for less money. I have found its too good to be a backup. It's different than the sound of my primary system but also good. BTW, I run it with Kef R300 monitors but it also sounded good with my Ohm Walsh speakers and my Spendor S3/5s.

Another vote for the Rogue Sphinx V2. Has a great built in phono preamp and will drive almost any speakers. The sound can also be somewhat tuned by swapping tubes. I think they might be on sale at Needle Dr.

I recently purchased a NuPrime IDA-8 for $1,000 and use it on my desk system. I find it superb for the money with an excellent onboard DAC though no phono or phones. Has Bluetooth as well if you want to use your phone and the sound is good - for Bluetooth. It's one of five amps I own and by far the cheapest but I find it well worth the pesos (and it's quite small).