Practice Time:
There is no ideal length of time for practising, especially for practising any one technique. However, an hour does seem like quite a long time for a beginner to be playing chords. It is quite normal for the ends of your fingers to become a little sore while playing, as you play more your fingers will become stronger and the skin on your finger tips will become tougher. However, if you are getting a lot of pain, particularly if it is in your wrist, arm or shoulder I would definitely play a little less.
Whether you are getting a lot of pain or just a little soreness, I would still take regular breaks while practising. This doesn't have to be long (not even a real break from your practice), but just half a minute or so between exercises to give your fingers a rest. This can also help to make your practice more relaxed and help you to sustain concentration.
Generally, it is better to practise for a shorter amount of time very regularly, than for a long time only now and then. For instance, 30mins every day is better than an hour three times a week.
Technique:
You are right that you don't want to move your fingers away from the strings while changing between chords. You keep them close, so that the chord change can be quicker and more accurate. There are two ways to achieve this:
- one is to keep any notes that are common to two chords in position while changing (for instance, keeping finger 2 on fret 2 string 4 when changing from Em to C);
- the second is to allow fingers to move along the string if the same finger is used on different frets in two chords (for instance, moving finger 1 along string 3 when changing from an E chord to a D chord. I call this technique, using a guide.
If you are planning to play chromatic chord change exercises (eg. A - A# - B etc.) you will probably want to use the same shape for each chord, so you would use the second technique, allowing each finger to move along the string it is on. However, you do not need to keep each finger fully pressed down to use this technique. Instead, you should lift enough so that the string is no longer pressed against the frets, but so that your finger is still touching the string. This makes a lot of difference. It allows you to keep the accuracy and speed of chord change, by keeping your fingers on the strings as "guides", but gives your fingers and wrist a little rest from pressing hard in between every chord change.
Use Easier Chords:
All chords on the guitar can be played with a wide variety of shapes, some easier to play than others. You can build up to playing more difficult chords by starting with simpler ones. For instance, start by playing major chord shapes only using three strings, before moving on to shapes with four, five and six strings. Below are some suggestions for how to play the chords A - A# - B, starting with easier three-string shapes:

As you can see each progression of chord changes uses the same shape, moving it up one fret at a time. (Sorry the chord boxes are a bit untidy - I scribbled them in a hurry!) The first shape will help you to move onto the fourth shape. Also, you could try the first shape with fingers 1, 2 & 3, if you find it difficult to press down two strings with finger 1. The third shape could be played with just one finger across the three strings.