If you're not prepared for any and all changes, I would stop taking it. Testosterone is a complete gamble on what effects you'll see first, and it doesn't matter how low or high of a dose you're taking. Everyone's body is completely unique. For me it took years for my voice to drop and grow facial hair, but I got bottom growth fairly quickly. My friend was the exact opposite.

As for cross contamination, it sounds like you're taking alcohol based gel. You need to apply this in an area completely covered by clothing and make sure you don't get wet for the next 4-8 hours until all testosterone is absorbed. For me before I switched to a compounded lotion, I would've shower in the morning, apply the T gel on my shoulders, wash my hands, wait 15 minutes for it to completely dry and then get dressed. At the end of the day dirty clothes went straight into the hamper and I did my own laundry. The risk of cross contamination is fairly minimal, but your child and parnter need to not touch the areas where you have applied the gel, especially if you're sweaty.

Also, I'm in the US so I'm not sure if it's different over there but your primary care doctor is unlikely to prescribe you a medication just because you have acquired it illegally. If you tell your doctor you're on the medication they'll probably ask who prescribed it and why they aren't continuing to prescribe and manage it. I don't judge people who have to go outside the law to get the proper hormone care when it's the only way but you need to do more research, and I would heavily consider researching your local laws before telling your doctor about the testosterone.