Overall, good thinking. Thanks for posting this.
I have a couple questions/clarifications.
I'm not sure what you mean by this ^.
Assuming you mean that your level of exhaustion decreases by 1, 'remove' might be a better word than 'regain'. ('Regain'
could be interpreted as level of exhaustion increases by 1.)
It could just be me, but I had to read this several times before I understood it. I found the way you were using 'base DC' confusing. I think it might be clearer to say this (assuming that I have understood correctly):
The DCs for recovering short and long rest abilities are reset to the base DCs. For each day since your last full day rest, add 1 to each DC.
Or maybe this (the longer I think, the more I edit; sorry

):
Recovering Short and Long Rest Abilities
You may recover all short rest abilities or all long rest abilities at any time. However, when you do so you must make an exhaustion check. You make a separate check after recovering short rest and long rest abilities. Each creature recovers abilities and make exhaustion checks separately.
The exhaustion check is an ability check against your primary class ability (Dexterity for rogues, Charisma for Sorcerers, etc.). If you fail the exhaustion check, you suffer 1 level of exhaustion if you recovered short rest abilities, or 3 levels of exhaustion if you recovered long rest abilities.
As noted below, immediately after a long rest, the DCs for exhaustion checks after recovering short rest and long rest abilities are 5 and 15, respectively. Following each exhaustion check, the corresponding DC increases by 5.
Travel Rest
8 hours long. One per day. You regain 1/2 of your Hit Dice. You remove 1 level of exhaustion. The DC for your exhaustion check after recovering short rest abilities is set to 5 plus 1 for each day since your long rest, and your next such check is made with advantage. The DC for your exhaustion check after recovering long rest abilities is set to 15 plus 1 for each day since your last long rest and your next such check is made with advantage.
Long Rest
24 hours of uninterrupted rest. You regain all hit points. You regain all Hit Dice. You remove all levels of exhaustion. You regain all abilities. The DCs for exhaustion checks after recovering short and long rest abilities are set to 5 and 15, respectively.