Don't forget that your budget needs to include any of the following which are applicable to your situation: storage fees for the TT, insurance for the TT, and TT vehicle license fees. These will be charged whether you just park the TT or go out often and use it.
As to your question of what to look for in a TT for small children: safe sleeping space (bunk rails?), safe stairs and ladders.
I've camped extensively with children from my own 6-week-old infants to my currently adult grandchildren. At least until kids approach the adult stage, plan on extra parental vigilance. There are so many wonderful unfamiliar things and places that they will want to investigate. No leaving them alone in the TT for even a short time. No "I thought you were watching them" situations. Watch out for camp area traffic, wild animals, uncontrolled tame animals, etc.
Have a great camping life! You'll be building many memories with your children. Don't wait too long to get started on this.