Can I add a subpanel?
One of the pros will give a long detailed answer, I'm sure.

But the short answer (from reading all their long detailed answers over many months) is that provided the existing panel is in decent shape, you can replace an existing double-pole breaker with a bigger breaker for a subpanel and use that subpanel to power several circuits, as you described.

However, you should put in a really big panel - i.e., big enough that it can eventually be your new main panel. A big (like 42 space) panel might cost you an extra $100 or $150 now compared to a small panel, but will allow you to gradually move circuits over time. Moving all those circuits is the big cost, particularly if you use the opportunity to add GFCI and/or AFCI as appropriate. But this way you don't have to spend it all at once. Then once you have everything moved you make the subpanel into your main panel, add a neutral-ground bond, and get rid of the old main panel.

Since you said "I am maxed out on breakers, as every 110v breaker is now a tandem breaker.", there is no way you can add GFCI or AFCI breakers right now. While you can add GFCI at the first receptacle in each chain and get full protection, and you can add AFCI in a separate box, it is a lot easier to put in breakers. Code generally allows you to keep the old stuff as-is, but for safety's sake, as well as if you do other upgrades (e.g., bathroom or kitchen remodel), you may want and/or need to make those upgrades at some time. With a new larger (sub)panel, you can do that easily. If the existing cables won't quite reach, you can splice them inside the existing panel, with that panel eventually becoming a large junction box when the the subpanel becomes your main panel.