Going cold turkey isn't the only way. For some people it works best, and most people probably see more results faster (which can be very motivating for some people), but if it's too difficult to stick to it isn't a help, and there's a tendency to throw everything out the window and go back to total SAD.
"You" refers to whoever is making the dietary changes, not the OP specifically, since the OP is not asking for him/herself. These suggestions are based on my own experiences, as well as watching friends try to go paleo and get overwhelmed.
A gradual approach can be a much easier adjustment psychologically.
As more foods are eliminated, be sure adequate fat is included to encourage satiety.
Step 1: If necessary, learn to cook, and prepare more meals at home. Switch to paleo-appropriate oils when cooking at home. Integrate higher quality animal products and produce to the greatest extent that budget allows. Focus on low glycemic index fruits and veggies, but integrating some sort of real produce with each meal is a positive step if it isn't happening already. Use grains and legumes that have been properly prepared (soaking and so forth) WAPF style for all homecooked meals. These steps should be pretty painless emotionally - you're not eliminating anything, just laying a foundation for future changes.
Step 2: Eliminate gluten. Preferably don't use gluten-free replacements (especially processed) unless they're paleo.
Step 3: Eat meals at least several times a week that don't include grains, legumes, and/or dairy. Not all of them, but enough to psychologically adjust to the idea of meals without these things. Use only high-quality, full-fat, preferably fermented dairy products, and experiment with using non-dairy milks for some purposes.
Step 4: Eliminate grains and legumes.
Step 5: Eliminate processed sugar. Don't replace them with large amounts of other sweetener, whether natural or artificial.
Step 6: Go low carb if not seeing adequate results from changes thus far.
Step 7: Experiment with eliminating dairy, going strict Whole30, and so forth.
These steps aren't set in stone - they should be adjusted according to what the individual in question finds easiest to eliminate. If cutting dairy is easy for the person, then that's a good earlier step. If keeping dairy in the diet helps with compliance to the rest of the diet, keep it in until other addictions are dealt with.
Alternatively, do this backwards. Go strict paleo, then add stuff back in as needed until you reach a point where you feel you can handle it rather than going back to complete SAD. Then work towards gradual elimination of those crutches.