Hello there and welcome to the site, you will indeed find good information and answers to your question.
I am a retired Marine, and military Air Traffic Controller that now works DoD for the USAF as a Tower Watch Supervisor here in FL.
So far it would seem that you are getting a lot of good advice and information in response to your question, and after reading everything, it would appear that none of it is really wrong.
First in your case, as a single mother, shift work is very demanding and it does help if you have a good support network and/or family near by, but by the same token it is feasible to do, and know a few single parents that seem to manage well. It is all about time management and priorities. Also, as your child gets older it will get easier on a few levels.
Second, if you are thinking of a career in ATC, as in something you can get into and get a retirement out of you are on the right track with the CTI program. In your case it will be an age thing whether you stick w/ the CTI or go w/ the ANG offer. Staying with the CTI program gives you the most time and flexibility to pursue the FAA. If you go ANG, and have to deploy, everything stops for you and your ability to pursue your dream is put on hold. The recruiters will tell you that it is only one weekend a month and 2 weeks out of the summer, but from my experience, the Guard and the Reserves are used heavily and are deployed often from 4 months anywhere up to a year.
Also, if you go into the Guard, you will go to the USAF school and then back to your Guard base to train, and after that you will do your Guard time only, which makes it hard to keep up proficiency unless you are willing to request extra guard time or deploy a lot. So without any active duty time and multiple facilities you would be hard pressed to get anyone to look at your resume and take it seriously as far as experience when looking for a job w/ FAA or contract tower. Not saying it isn't possible, just harder. You would have a shot w/ a contract tower, if you had a CTO and were willing to go to a location that historically, they have a hard time staffing or keeping staffed. With the FAA it would be the time thing.
I would recommend staying with the CTI, and applying with the FAA via every available means including PUBNATs..... If you so approach the age cutoff for the FAA then go and apply for the ANG and get the training and whatever experience you can get. You may be fortunate to land a full time technical guard position which is the best of both worlds, civilian and military. You get paid as a GS employee w/ federal benefits, except for the 2 days a month and 2 weeks a year, when you are doing ANG duties at which time you are paid military salary and benefits. You can still deploy, and get all the extra schools and training the military have to offer.
If you do go the military route, after your enlistment is up you can look into getting a DoD job as a civilian controller as their age limit is 36 as long as you have military experience which you would have.
I know it is a lot of information, but feel free to ask the questions, more than once if you have to. The main thing is to weed through everything and find out what exactly applies to you and best fits your circumstances as everyone's are different.