At least one positive change is that you'll feel less fragile. Which is what you're going for, so that you'll get for sure. Fights will drag a bit, that's also for sure. If you want to know by how much, check with D&D 4e players, because it had pretty much this rule and fights were long.
As for problems you'll run into a low level; this gives all creatures a variable bump to their HP, but the bump is bigger the lower their original HP was. This will change the challenge rating of a lot of creatures. Things that used to be "a nuisance" might become incredibly dangerous suddenly.
- a Zombie (3d8 with con 16) goes from 22 hp to 35 (about 50% more)
- a Hobgoblin (2d8 with con 12) goes from 11 hp to 22 (twice as many)
- a Goblin (2d6 with con 10) goes from 7 hp to 17 (over twice as m any)
- but a Stirge (1d4 with con 11) goes from 2 hp to 13 (6 times more)
So don't assume challenge ratings will stay the same.
Battles with groups of small creatures will be much more dangerous because their effective HP goes up a ton compared to a single larger creature.
Another thing that happens (which we've also seen in D&D 4e) is that because HP scaled but damage remains the same, is that once-powerful abilities will feel less powerful. This especially hurts the limited-use ones.
A burning hands spell will normally almost certainly waste a group of Goblins; any that fail their save are dead. Maybe even the ones that succeed, if you roll well. This leaves such a spell as a powerful and cool ability that you can use when outnumbered. With your change, you'd have to roll maximum damage and they'd need to fail a save to kill anything.
Using Burning Hands won't be a cool encounter-changing effect, but just another spell to injure a few opponents. And you still only get about 2 spells per day at 1st level. Likewise, a spell like Sleep becomes almost useless, especially against groups. You might be able to drop a single higher level target, but definitely not a group of Goblins.
A lot of the times using a Daily ability in 4e (the equivalent of your highest level spell-slot in 5e) would feel sort of "meh" because while it did some more damage than a lower level ability, it did not have a noticable impact on the battlefield. In 5e, it currently does tend to have a serious impact, but your change will likely remove much of it.