Hi Anna and welcome to Chordie,
Keeping the timing constant and changing chords are the two most difficult things that most students have to master and they are generally linked.
What usually happens is that because the chord is not changed fast enough, or fumbled , the strumming hand slows down to let the chord change hand catch up and then we are behind the tempo of the song. The reason that for a right handed person the strumming hand is the right hand is that this one has the most important job - keeping time and rhythm. You must concentrate on this - the left hand then has to catch up. It is not and easy concept as the natural reaction is to correct the chord so it takes some working on.
To demonstrate this I often play a fast rock song for my students and then slow it down so that they can see that every often there is a strum on all open strings between chord changes but because the tempo is regular it is not noticed.
I have a seven year old student who has been coming to me for a year and in May she played 2 songs at my 65th birthday party in front of my friends, other students, her parents and a restaurent full of people. Her co-ordination is typical of a 7 year old and she fumbles some chords but because she can keep her right hand in strict tempo she pulled it off. She did a great job and people were amazed at her performance, I noticed her mistakes but no one else did .
When practicing, the use of a metronome, a backing track, an original recording (if the key is the same) or playing with someone who can keep time will all help.
Keep at it it will come together in time and with practice.